Theorem wessf1ornlem 45632

Description: Given a function 𝐹 on a well-ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
wessf1ornlem.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
wessf1ornlem.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
wessf1ornlem.r	⊢ (𝜑 → 𝑅 We 𝐴)
wessf1ornlem.g	⊢ 𝐺 = (𝑦 ∈ ran 𝐹 ↦ (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥))

Assertion

Ref	Expression
wessf1ornlem	⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem wessf1ornlem

Dummy variables 𝑡 𝑢 𝑣 𝑤 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wessf1ornlem.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		cnvimass 6034	. . . . . . . 8 ⊢ (◡𝐹 “ {𝑢}) ⊆ dom 𝐹
3		wessf1ornlem.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4	3	fndmd 6590	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → dom 𝐹 = 𝐴)
6	2, 5	sseqtrid 3957	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (◡𝐹 “ {𝑢}) ⊆ 𝐴)
7		wessf1ornlem.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 We 𝐴)
8	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → 𝑅 We 𝐴)
9	2, 4	sseqtrid 3957	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐹 “ {𝑢}) ⊆ 𝐴)
10	1, 9	ssexd 5252	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐹 “ {𝑢}) ∈ V)
11	10	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (◡𝐹 “ {𝑢}) ∈ V)
12		inisegn0 6050	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑢}) ≠ ∅)
13	12	bilani 505	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (◡𝐹 “ {𝑢}) ≠ ∅)
14		wereu 5614	. . . . . . . . 9 ⊢ ((𝑅 We 𝐴 ∧ ((◡𝐹 “ {𝑢}) ∈ V ∧ (◡𝐹 “ {𝑢}) ⊆ 𝐴 ∧ (◡𝐹 “ {𝑢}) ≠ ∅)) → ∃!𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
15	8, 11, 6, 13, 14	syl13anc 1380	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ∃!𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
16		riotacl 7330	. . . . . . . 8 ⊢ (∃!𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣 → (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ (◡𝐹 “ {𝑢}))
17	15, 16	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ (◡𝐹 “ {𝑢}))
18	6, 17	sseldd 3916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ 𝐴)
19	18	ralrimiva 3131	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ ran 𝐹(℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ 𝐴)
20		wessf1ornlem.g	. . . . . . 7 ⊢ 𝐺 = (𝑦 ∈ ran 𝐹 ↦ (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥))
21		sneq 4565	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑢 → {𝑦} = {𝑢})
22	21	imaeq2d 6012	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑢}))
23	22	raleqdv 3297	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → (∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥 ↔ ∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥))
24	22, 23	riotaeqbidv 7316	. . . . . . . . 9 ⊢ (𝑦 = 𝑢 → (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥) = (℩𝑥 ∈ (◡𝐹 “ {𝑢})∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥))
25		breq1 5075	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑡 → (𝑧𝑅𝑥 ↔ 𝑡𝑅𝑥))
26	25	notbid 319	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑡𝑅𝑥))
27	26	cbvralvw 3217	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥 ↔ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑥)
28		breq2 5076	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑣 → (𝑡𝑅𝑥 ↔ 𝑡𝑅𝑣))
29	28	notbid 319	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑣 → (¬ 𝑡𝑅𝑥 ↔ ¬ 𝑡𝑅𝑣))
30	29	ralbidv 3162	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑥 ↔ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
31	27, 30	bitrid 284	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥 ↔ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
32	31	cbvriotavw 7323	. . . . . . . . 9 ⊢ (℩𝑥 ∈ (◡𝐹 “ {𝑢})∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥) = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
33	24, 32	eqtrdi 2790	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥) = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
34	33	cbvmptv 5176	. . . . . . 7 ⊢ (𝑦 ∈ ran 𝐹 ↦ (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥)) = (𝑢 ∈ ran 𝐹 ↦ (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
35	20, 34	eqtri 2762	. . . . . 6 ⊢ 𝐺 = (𝑢 ∈ ran 𝐹 ↦ (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
36	35	rnmptss 7064	. . . . 5 ⊢ (∀𝑢 ∈ ran 𝐹(℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ 𝐴 → ran 𝐺 ⊆ 𝐴)
37	19, 36	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐴)
38	1, 37	sselpwd 5256	. . 3 ⊢ (𝜑 → ran 𝐺 ∈ 𝒫 𝐴)
39		dffn3 6667	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
40	3, 39	sylib 219	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
41	40, 37	fssresd 6694	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺⟶ran 𝐹)
42		fvres 6846	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ran 𝐺 → ((𝐹 ↾ ran 𝐺)‘𝑤) = (𝐹‘𝑤))
43	42	eqcomd 2745	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ran 𝐺 → (𝐹‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑤))
44	43	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → (𝐹‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑤))
45		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡))
46		fvres 6846	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ran 𝐺 → ((𝐹 ↾ ran 𝐺)‘𝑡) = (𝐹‘𝑡))
47	46	ad2antlr 733	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → ((𝐹 ↾ ran 𝐺)‘𝑡) = (𝐹‘𝑡))
48	44, 45, 47	3eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → (𝐹‘𝑤) = (𝐹‘𝑡))
49	48	3adantl1 1173	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → (𝐹‘𝑤) = (𝐹‘𝑡))
50		simpl1 1198	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝜑)
51		simpl3 1200	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑡 ∈ ran 𝐺)
52		simpl2 1199	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑤 ∈ ran 𝐺)
53		id 22	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑤) = (𝐹‘𝑡) → (𝐹‘𝑤) = (𝐹‘𝑡))
54	53	eqcomd 2745	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑤) = (𝐹‘𝑡) → (𝐹‘𝑡) = (𝐹‘𝑤))
55	54	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → (𝐹‘𝑡) = (𝐹‘𝑤))
56		eleq1w 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → (𝑏 ∈ ran 𝐺 ↔ 𝑤 ∈ ran 𝐺))
57	56	3anbi3d 1450	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → ((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ↔ (𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑤 ∈ ran 𝐺)))
58		fveq2 6827	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → (𝐹‘𝑏) = (𝐹‘𝑤))
59	58	eqeq2d 2750	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → ((𝐹‘𝑡) = (𝐹‘𝑏) ↔ (𝐹‘𝑡) = (𝐹‘𝑤)))
60	57, 59	anbi12d 638	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑤 → (((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑏)) ↔ ((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑤 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑤))))
61		breq1 5075	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → (𝑏𝑅𝑡 ↔ 𝑤𝑅𝑡))
62	61	notbid 319	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑤 → (¬ 𝑏𝑅𝑡 ↔ ¬ 𝑤𝑅𝑡))
63	60, 62	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑤 → ((((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑏)) → ¬ 𝑏𝑅𝑡) ↔ (((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑤 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑤)) → ¬ 𝑤𝑅𝑡)))
64		eleq1w 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑡 → (𝑎 ∈ ran 𝐺 ↔ 𝑡 ∈ ran 𝐺))
65	64	3anbi2d 1449	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑡 → ((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ↔ (𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺)))
66		fveqeq2 6836	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑡 → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (𝐹‘𝑡) = (𝐹‘𝑏)))
67	65, 66	anbi12d 638	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑡 → (((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ↔ ((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑏))))
68		breq2 5076	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑡 → (𝑏𝑅𝑎 ↔ 𝑏𝑅𝑡))
69	68	notbid 319	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑡 → (¬ 𝑏𝑅𝑎 ↔ ¬ 𝑏𝑅𝑡))
70	67, 69	imbi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑡 → ((((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ¬ 𝑏𝑅𝑎) ↔ (((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑏)) → ¬ 𝑏𝑅𝑡)))
71		eleq1w 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑏 → (𝑡 ∈ ran 𝐺 ↔ 𝑏 ∈ ran 𝐺))
72	71	3anbi3d 1450	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑏 → ((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ↔ (𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺)))
73		fveq2 6827	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑏 → (𝐹‘𝑡) = (𝐹‘𝑏))
74	73	eqeq2d 2750	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑏 → ((𝐹‘𝑎) = (𝐹‘𝑡) ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
75	72, 74	anbi12d 638	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑏 → (((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑡)) ↔ ((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑏))))
76		breq1 5075	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑏 → (𝑡𝑅𝑎 ↔ 𝑏𝑅𝑎))
77	76	notbid 319	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑏 → (¬ 𝑡𝑅𝑎 ↔ ¬ 𝑏𝑅𝑎))
78	75, 77	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑏 → ((((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑡)) → ¬ 𝑡𝑅𝑎) ↔ (((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ¬ 𝑏𝑅𝑎)))
79		eleq1w 2822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑎 → (𝑤 ∈ ran 𝐺 ↔ 𝑎 ∈ ran 𝐺))
80	79	3anbi2d 1449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑎 → ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ↔ (𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺)))
81		fveqeq2 6836	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑎 → ((𝐹‘𝑤) = (𝐹‘𝑡) ↔ (𝐹‘𝑎) = (𝐹‘𝑡)))
82	80, 81	anbi12d 638	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑎 → (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ↔ ((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑡))))
83		breq2 5076	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑎 → (𝑡𝑅𝑤 ↔ 𝑡𝑅𝑎))
84	83	notbid 319	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑎 → (¬ 𝑡𝑅𝑤 ↔ ¬ 𝑡𝑅𝑎))
85	82, 84	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑎 → ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → ¬ 𝑡𝑅𝑤) ↔ (((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑡)) → ¬ 𝑡𝑅𝑎)))
86	35	elrnmpt 5900	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran 𝐺 ↔ ∃𝑢 ∈ ran 𝐹 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)))
87	86	elv 3436	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ran 𝐺 ↔ ∃𝑢 ∈ ran 𝐹 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
88	87	birani 504	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ran 𝐺 ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → ∃𝑢 ∈ ran 𝐹 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
89	88	3ad2antl2 1193	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → ∃𝑢 ∈ ran 𝐹 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
90		simp3 1144	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
91	90	eqcomd 2745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) = 𝑤)
92		simp11 1210	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝜑)
93		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
94		breq2 5076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑣 = 𝑤 → (𝑡𝑅𝑣 ↔ 𝑡𝑅𝑤))
95	94	notbid 319	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 = 𝑤 → (¬ 𝑡𝑅𝑣 ↔ ¬ 𝑡𝑅𝑤))
96	95	ralbidv 3162	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣 = 𝑤 → (∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣 ↔ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤))
97	96	cbvriotavw 7323	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) = (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
98	93, 97	eqtr2di 2791	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤)
99	98	3ad2ant3 1141	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤)
100	96	cbvreuvw 3366	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃!𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣 ↔ ∃!𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
101	15, 100	sylib 219	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ∃!𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
102		riota1 7334	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃!𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤 → ((𝑤 ∈ (◡𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) ↔ (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤))
103	101, 102	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ((𝑤 ∈ (◡𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) ↔ (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤))
104	103	3adant3 1138	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ((𝑤 ∈ (◡𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) ↔ (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤))
105	99, 104	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑤 ∈ (◡𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤))
106	105	simpld 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑤 ∈ (◡𝐹 “ {𝑢}))
107	92, 106	syld3an1 1418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑤 ∈ (◡𝐹 “ {𝑢}))
108		simp2 1143	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑢 ∈ ran 𝐹)
109	92, 108, 15	syl2anc 590	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ∃!𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
110	96	riota2 7338	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ (◡𝐹 “ {𝑢}) ∧ ∃!𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → (∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤 ↔ (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) = 𝑤))
111	107, 109, 110	syl2anc 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤 ↔ (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) = 𝑤))
112	91, 111	mpbird 258	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
113	112	3adant1r 1184	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
114	37	sselda 3915	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐺) → 𝑡 ∈ 𝐴)
115	114	3adant2 1137	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) → 𝑡 ∈ 𝐴)
116	115	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑡 ∈ 𝐴)
117	116	3ad2ant1 1139	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑡 ∈ 𝐴)
118	54	ad2antlr 733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹) → (𝐹‘𝑡) = (𝐹‘𝑤))
119	118	3adant3 1138	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝐹‘𝑡) = (𝐹‘𝑤))
120		fniniseg 7001	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢)))
121	92, 3, 120	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑤 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢)))
122	107, 121	mpbid 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢))
123	122	simprd 496	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝐹‘𝑤) = 𝑢)
124	123	3adant1r 1184	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝐹‘𝑤) = 𝑢)
125	119, 124	eqtrd 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝐹‘𝑡) = 𝑢)
126		fniniseg 7001	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Fn 𝐴 → (𝑡 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑡 ∈ 𝐴 ∧ (𝐹‘𝑡) = 𝑢)))
127	3, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑡 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑡 ∈ 𝐴 ∧ (𝐹‘𝑡) = 𝑢)))
128	127	3ad2ant1 1139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) → (𝑡 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑡 ∈ 𝐴 ∧ (𝐹‘𝑡) = 𝑢)))
129	128	ad2antrr 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹) → (𝑡 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑡 ∈ 𝐴 ∧ (𝐹‘𝑡) = 𝑢)))
130	129	3adant3 1138	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑡 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑡 ∈ 𝐴 ∧ (𝐹‘𝑡) = 𝑢)))
131	117, 125, 130	mpbir2and 719	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑡 ∈ (◡𝐹 “ {𝑢}))
132		rspa 3228	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤 ∧ 𝑡 ∈ (◡𝐹 “ {𝑢})) → ¬ 𝑡𝑅𝑤)
133	113, 131, 132	syl2anc 590	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ¬ 𝑡𝑅𝑤)
134	133	rexlimdv3a 3144	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → (∃𝑢 ∈ ran 𝐹 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → ¬ 𝑡𝑅𝑤))
135	89, 134	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → ¬ 𝑡𝑅𝑤)
136	85, 135	chvarvv 1996	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑡)) → ¬ 𝑡𝑅𝑎)
137	78, 136	chvarvv 1996	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ¬ 𝑏𝑅𝑎)
138	70, 137	chvarvv 1996	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑏)) → ¬ 𝑏𝑅𝑡)
139	63, 138	chvarvv 1996	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑤 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑤)) → ¬ 𝑤𝑅𝑡)
140	50, 51, 52, 55, 139	syl31anc 1381	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → ¬ 𝑤𝑅𝑡)
141		weso 5609	. . . . . . . . . . . . . 14 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
142	7, 141	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 Or 𝐴)
143	142	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑅 Or 𝐴)
144	143	3ad2antl1 1192	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑅 Or 𝐴)
145	37	sselda 3915	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺) → 𝑤 ∈ 𝐴)
146	145	3adant3 1138	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) → 𝑤 ∈ 𝐴)
147	146	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑤 ∈ 𝐴)
148		sotrieq2 5558	. . . . . . . . . . 11 ⊢ ((𝑅 Or 𝐴 ∧ (𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴)) → (𝑤 = 𝑡 ↔ (¬ 𝑤𝑅𝑡 ∧ ¬ 𝑡𝑅𝑤)))
149	144, 147, 116, 148	syl12anc 842	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → (𝑤 = 𝑡 ↔ (¬ 𝑤𝑅𝑡 ∧ ¬ 𝑡𝑅𝑤)))
150	140, 135, 149	mpbir2and 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑤 = 𝑡)
151	49, 150	syldan 597	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → 𝑤 = 𝑡)
152	151	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) → (((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡))
153	152	3expb 1126	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺)) → (((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡))
154	153	ralrimivva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑤 ∈ ran 𝐺∀𝑡 ∈ ran 𝐺(((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡))
155		dff13 7198	. . . . 5 ⊢ ((𝐹 ↾ ran 𝐺):ran 𝐺–1-1→ran 𝐹 ↔ ((𝐹 ↾ ran 𝐺):ran 𝐺⟶ran 𝐹 ∧ ∀𝑤 ∈ ran 𝐺∀𝑡 ∈ ran 𝐺(((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡)))
156	41, 154, 155	sylanbrc 589	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺–1-1→ran 𝐹)
157		riotaex 7317	. . . . . . . . . . 11 ⊢ (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V
158	157	rgenw 3057	. . . . . . . . . 10 ⊢ ∀𝑢 ∈ ran 𝐹(℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V
159	35	fnmpt 6625	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ ran 𝐹(℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V → 𝐺 Fn ran 𝐹)
160	158, 159	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn ran 𝐹)
161		dffn3 6667	. . . . . . . . 9 ⊢ (𝐺 Fn ran 𝐹 ↔ 𝐺:ran 𝐹⟶ran 𝐺)
162	160, 161	sylib 219	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ran 𝐹⟶ran 𝐺)
163	162	ffvelcdmda 7025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (𝐺‘𝑢) ∈ ran 𝐺)
164	163	fvresd 6847	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ((𝐹 ↾ ran 𝐺)‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝑢)))
165		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 ∈ ran 𝐹)
166	157	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V)
167	20, 33, 165, 166	fvmptd3 6959	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (𝐺‘𝑢) = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
168	167, 17	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢}))
169		fvex 6840	. . . . . . . . . . . 12 ⊢ (𝐺‘𝑢) ∈ V
170		eleq1 2827	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐺‘𝑢) → (𝑤 ∈ (◡𝐹 “ {𝑢}) ↔ (𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢})))
171		eleq1 2827	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐺‘𝑢) → (𝑤 ∈ 𝐴 ↔ (𝐺‘𝑢) ∈ 𝐴))
172		fveqeq2 6836	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐺‘𝑢) → ((𝐹‘𝑤) = 𝑢 ↔ (𝐹‘(𝐺‘𝑢)) = 𝑢))
173	171, 172	anbi12d 638	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐺‘𝑢) → ((𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢) ↔ ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢)))
174	170, 173	bibi12d 346	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐺‘𝑢) → ((𝑤 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢)) ↔ ((𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢}) ↔ ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢))))
175	174	imbi2d 341	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐺‘𝑢) → ((𝜑 → (𝑤 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢))) ↔ (𝜑 → ((𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢}) ↔ ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢)))))
176	3, 120	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑤 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢)))
177	169, 175, 176	vtocl 3503	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢}) ↔ ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢)))
178	177	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ((𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢}) ↔ ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢)))
179	168, 178	mpbid 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢))
180	179	simprd 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (𝐹‘(𝐺‘𝑢)) = 𝑢)
181	164, 180	eqtr2d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 = ((𝐹 ↾ ran 𝐺)‘(𝐺‘𝑢)))
182		fveq2 6827	. . . . . . . 8 ⊢ (𝑤 = (𝐺‘𝑢) → ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘(𝐺‘𝑢)))
183	182	rspceeqv 3583	. . . . . . 7 ⊢ (((𝐺‘𝑢) ∈ ran 𝐺 ∧ 𝑢 = ((𝐹 ↾ ran 𝐺)‘(𝐺‘𝑢))) → ∃𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤))
184	163, 181, 183	syl2anc 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ∃𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤))
185	184	ralrimiva 3131	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ ran 𝐹∃𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤))
186		dffo3 7043	. . . . 5 ⊢ ((𝐹 ↾ ran 𝐺):ran 𝐺–onto→ran 𝐹 ↔ ((𝐹 ↾ ran 𝐺):ran 𝐺⟶ran 𝐹 ∧ ∀𝑢 ∈ ran 𝐹∃𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤)))
187	41, 185, 186	sylanbrc 589	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺–onto→ran 𝐹)
188		df-f1o 6492	. . . 4 ⊢ ((𝐹 ↾ ran 𝐺):ran 𝐺–1-1-onto→ran 𝐹 ↔ ((𝐹 ↾ ran 𝐺):ran 𝐺–1-1→ran 𝐹 ∧ (𝐹 ↾ ran 𝐺):ran 𝐺–onto→ran 𝐹))
189	156, 187, 188	sylanbrc 589	. . 3 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺–1-1-onto→ran 𝐹)
190		reseq2 5926	. . . . 5 ⊢ (𝑣 = ran 𝐺 → (𝐹 ↾ 𝑣) = (𝐹 ↾ ran 𝐺))
191		id 22	. . . . 5 ⊢ (𝑣 = ran 𝐺 → 𝑣 = ran 𝐺)
192		eqidd 2740	. . . . 5 ⊢ (𝑣 = ran 𝐺 → ran 𝐹 = ran 𝐹)
193	190, 191, 192	f1oeq123d 6761	. . . 4 ⊢ (𝑣 = ran 𝐺 → ((𝐹 ↾ 𝑣):𝑣–1-1-onto→ran 𝐹 ↔ (𝐹 ↾ ran 𝐺):ran 𝐺–1-1-onto→ran 𝐹))
194	193	rspcev 3560	. . 3 ⊢ ((ran 𝐺 ∈ 𝒫 𝐴 ∧ (𝐹 ↾ ran 𝐺):ran 𝐺–1-1-onto→ran 𝐹) → ∃𝑣 ∈ 𝒫 𝐴(𝐹 ↾ 𝑣):𝑣–1-1-onto→ran 𝐹)
195	38, 189, 194	syl2anc 590	. 2 ⊢ (𝜑 → ∃𝑣 ∈ 𝒫 𝐴(𝐹 ↾ 𝑣):𝑣–1-1-onto→ran 𝐹)
196		reseq2 5926	. . . 4 ⊢ (𝑣 = 𝑥 → (𝐹 ↾ 𝑣) = (𝐹 ↾ 𝑥))
197		id 22	. . . 4 ⊢ (𝑣 = 𝑥 → 𝑣 = 𝑥)
198		eqidd 2740	. . . 4 ⊢ (𝑣 = 𝑥 → ran 𝐹 = ran 𝐹)
199	196, 197, 198	f1oeq123d 6761	. . 3 ⊢ (𝑣 = 𝑥 → ((𝐹 ↾ 𝑣):𝑣–1-1-onto→ran 𝐹 ↔ (𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹))
200	199	cbvrexvw 3218	. 2 ⊢ (∃𝑣 ∈ 𝒫 𝐴(𝐹 ↾ 𝑣):𝑣–1-1-onto→ran 𝐹 ↔ ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)
201	195, 200	sylib 219	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  ∃!wreu 3342  Vcvv 3431   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  {csn 4555   class class class wbr 5072   ↦ cmpt 5153   Or wor 5525   We wwe 5570  ^◡ccnv 5617  dom cdm 5618  ran crn 5619   ↾ cres 5620   “ cima 5621   Fn wfn 6480  ⟶wf 6481  –1-1→wf1 6482  –onto→wfo 6483  –1-1-onto→wf1o 6484  ‘cfv 6485  ℩crio 7312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313

This theorem is referenced by:  wessf1orn  45633