Theorem wessf1ornlem 45794

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 6085	. . . . . . . 8 ⊢ (◡𝐹 “ {𝑢}) ⊆ dom 𝐹
3		wessf1ornlem.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4	3	fndmd 6641	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → dom 𝐹 = 𝐴)
6	2, 5	sseqtrid 3987	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (◡𝐹 “ {𝑢}) ⊆ 𝐴)
7		wessf1ornlem.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 We 𝐴)
8	7	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → 𝑅 We 𝐴)
9	2, 4	sseqtrid 3987	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐹 “ {𝑢}) ⊆ 𝐴)
10	1, 9	ssexd 5295	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐹 “ {𝑢}) ∈ V)
11	10	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (◡𝐹 “ {𝑢}) ∈ V)
12		inisegn0 6101	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑢}) ≠ ∅)
13	12	bilani 509	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (◡𝐹 “ {𝑢}) ≠ ∅)
14		wereu 5658	. . . . . . . . 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 1397	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ∃!𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
16		riotacl 7385	. . . . . . . 8 ⊢ (∃!𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣 → (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ (◡𝐹 “ {𝑢}))
17	15, 16	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ (◡𝐹 “ {𝑢}))
18	6, 17	sseldd 3946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ 𝐴)
19	18	ralrimiva 3163	. . . . 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 4604	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑢 → {𝑦} = {𝑢})
22	21	imaeq2d 6063	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑢}))
23	22	raleqdv 3329	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → (∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥 ↔ ∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥))
24	22, 23	riotaeqbidv 7371	. . . . . . . . 9 ⊢ (𝑦 = 𝑢 → (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥) = (℩𝑥 ∈ (◡𝐹 “ {𝑢})∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥))
25		breq1 5116	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑡 → (𝑧𝑅𝑥 ↔ 𝑡𝑅𝑥))
26	25	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑡 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑡𝑅𝑥))
27	26	cbvralvw 3249	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥 ↔ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑥)
28		breq2 5117	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑣 → (𝑡𝑅𝑥 ↔ 𝑡𝑅𝑣))
29	28	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑣 → (¬ 𝑡𝑅𝑥 ↔ ¬ 𝑡𝑅𝑣))
30	29	ralbidv 3194	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑥 ↔ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
31	27, 30	bitrid 286	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥 ↔ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
32	31	cbvriotavw 7378	. . . . . . . . 9 ⊢ (℩𝑥 ∈ (◡𝐹 “ {𝑢})∀𝑧 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥) = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
33	24, 32	eqtrdi 2820	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥) = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
34	33	cbvmptv 5219	. . . . . . 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 2792	. . . . . 6 ⊢ 𝐺 = (𝑢 ∈ ran 𝐹 ↦ (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
36	35	rnmptss 7119	. . . . 5 ⊢ (∀𝑢 ∈ ran 𝐹(℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ 𝐴 → ran 𝐺 ⊆ 𝐴)
37	19, 36	syl 18	. . . 4 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐴)
38	1, 37	sselpwd 5299	. . 3 ⊢ (𝜑 → ran 𝐺 ∈ 𝒫 𝐴)
39		dffn3 6719	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
40	3, 39	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
41	40, 37	fssresd 6746	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺⟶ran 𝐹)
42		fvres 6901	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ran 𝐺 → ((𝐹 ↾ ran 𝐺)‘𝑤) = (𝐹‘𝑤))
43	42	eqcomd 2775	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ran 𝐺 → (𝐹‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑤))
44	43	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → (𝐹‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑤))
45		simpr 489	. . . . . . . . . . 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 6901	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ran 𝐺 → ((𝐹 ↾ ran 𝐺)‘𝑡) = (𝐹‘𝑡))
47	46	ad2antlr 739	. . . . . . . . . . 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 2808	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → (𝐹‘𝑤) = (𝐹‘𝑡))
49	48	3adantl1 1183	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → (𝐹‘𝑤) = (𝐹‘𝑡))
50		simpl1 1208	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝜑)
51		simpl3 1210	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑡 ∈ ran 𝐺)
52		simpl2 1209	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑤 ∈ ran 𝐺)
53		id 23	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑤) = (𝐹‘𝑡) → (𝐹‘𝑤) = (𝐹‘𝑡))
54	53	eqcomd 2775	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑤) = (𝐹‘𝑡) → (𝐹‘𝑡) = (𝐹‘𝑤))
55	54	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → (𝐹‘𝑡) = (𝐹‘𝑤))
56		eleq1w 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → (𝑏 ∈ ran 𝐺 ↔ 𝑤 ∈ ran 𝐺))
57	56	3anbi3d 1468	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → ((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ↔ (𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑤 ∈ ran 𝐺)))
58		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → (𝐹‘𝑏) = (𝐹‘𝑤))
59	58	eqeq2d 2780	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → ((𝐹‘𝑡) = (𝐹‘𝑏) ↔ (𝐹‘𝑡) = (𝐹‘𝑤)))
60	57, 59	anbi12d 643	. . . . . . . . . . . . 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 5116	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → (𝑏𝑅𝑡 ↔ 𝑤𝑅𝑡))
62	61	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑤 → (¬ 𝑏𝑅𝑡 ↔ ¬ 𝑤𝑅𝑡))
63	60, 62	imbi12d 347	. . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑡 → (𝑎 ∈ ran 𝐺 ↔ 𝑡 ∈ ran 𝐺))
65	64	3anbi2d 1467	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑡 → ((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ↔ (𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺)))
66		fveqeq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑡 → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (𝐹‘𝑡) = (𝐹‘𝑏)))
67	65, 66	anbi12d 643	. . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑡 → (𝑏𝑅𝑎 ↔ 𝑏𝑅𝑡))
69	68	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑡 → (¬ 𝑏𝑅𝑎 ↔ ¬ 𝑏𝑅𝑡))
70	67, 69	imbi12d 347	. . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑏 → (𝑡 ∈ ran 𝐺 ↔ 𝑏 ∈ ran 𝐺))
72	71	3anbi3d 1468	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑏 → ((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ↔ (𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺)))
73		fveq2 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑏 → (𝐹‘𝑡) = (𝐹‘𝑏))
74	73	eqeq2d 2780	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑏 → ((𝐹‘𝑎) = (𝐹‘𝑡) ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
75	72, 74	anbi12d 643	. . . . . . . . . . . . . . 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 5116	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑏 → (𝑡𝑅𝑎 ↔ 𝑏𝑅𝑎))
77	76	notbid 321	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑏 → (¬ 𝑡𝑅𝑎 ↔ ¬ 𝑏𝑅𝑎))
78	75, 77	imbi12d 347	. . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑎 → (𝑤 ∈ ran 𝐺 ↔ 𝑎 ∈ ran 𝐺))
80	79	3anbi2d 1467	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑎 → ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ↔ (𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺)))
81		fveqeq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑎 → ((𝐹‘𝑤) = (𝐹‘𝑡) ↔ (𝐹‘𝑎) = (𝐹‘𝑡)))
82	80, 81	anbi12d 643	. . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑎 → (𝑡𝑅𝑤 ↔ 𝑡𝑅𝑎))
84	83	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑎 → (¬ 𝑡𝑅𝑤 ↔ ¬ 𝑡𝑅𝑎))
85	82, 84	imbi12d 347	. . . . . . . . . . . . . . 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 5949	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran 𝐺 ↔ ∃𝑢 ∈ ran 𝐹 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)))
87	86	elv 3468	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ran 𝐺 ↔ ∃𝑢 ∈ ran 𝐹 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
88	87	birani 508	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ran 𝐺 ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → ∃𝑢 ∈ ran 𝐹 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
89	88	3ad2antl2 1203	. . . . . . . . . . . . . . . 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 1154	. . . . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝜑)
93		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
94		breq2 5117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑣 = 𝑤 → (𝑡𝑅𝑣 ↔ 𝑡𝑅𝑤))
95	94	notbid 321	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 = 𝑤 → (¬ 𝑡𝑅𝑣 ↔ ¬ 𝑡𝑅𝑤))
96	95	ralbidv 3194	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣 = 𝑤 → (∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣 ↔ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤))
97	96	cbvriotavw 7378	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) = (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
98	93, 97	eqtr2di 2821	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤)
99	98	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤)
100	96	cbvreuvw 3398	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃!𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣 ↔ ∃!𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
101	15, 100	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ∃!𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
102		riota1 7389	. . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ((𝑤 ∈ (◡𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) ↔ (℩𝑤 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤))
104	103	3adant3 1148	. . . . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑤 ∈ (◡𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤))
106	105	simpld 499	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑤 ∈ (◡𝐹 “ {𝑢}))
107	92, 106	syld3an1 1435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹 ∧ 𝑤 = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑤 ∈ (◡𝐹 “ {𝑢}))
108		simp2 1153	. . . . . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . . . . 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 7393	. . . . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐺) → 𝑡 ∈ 𝐴)
115	114	3adant2 1147	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) → 𝑡 ∈ 𝐴)
116	115	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑡 ∈ 𝐴)
117	116	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) ∧ 𝑢 ∈ ran 𝐹) → (𝐹‘𝑡) = (𝐹‘𝑤))
119	118	3adant3 1148	. . . . . . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢)))
121	92, 3, 120	3syl 19	. . . . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Fn 𝐴 → (𝑡 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑡 ∈ 𝐴 ∧ (𝐹‘𝑡) = 𝑢)))
127	3, 126	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑡 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑡 ∈ 𝐴 ∧ (𝐹‘𝑡) = 𝑢)))
128	127	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) → (𝑡 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑡 ∈ 𝐴 ∧ (𝐹‘𝑡) = 𝑢)))
129	128	ad2antrr 738	. . . . . . . . . . . . . . . . . . . 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 1148	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . 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 3260	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤 ∧ 𝑡 ∈ (◡𝐹 “ {𝑢})) → ¬ 𝑡𝑅𝑤)
133	113, 131, 132	syl2anc 595	. . . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → ¬ 𝑡𝑅𝑤)
136	85, 135	chvarvv 2016	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑡)) → ¬ 𝑡𝑅𝑎)
137	78, 136	chvarvv 2016	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → ¬ 𝑏𝑅𝑎)
138	70, 137	chvarvv 2016	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑏 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑏)) → ¬ 𝑏𝑅𝑡)
139	63, 138	chvarvv 2016	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ran 𝐺 ∧ 𝑤 ∈ ran 𝐺) ∧ (𝐹‘𝑡) = (𝐹‘𝑤)) → ¬ 𝑤𝑅𝑡)
140	50, 51, 52, 55, 139	syl31anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → ¬ 𝑤𝑅𝑡)
141		weso 5653	. . . . . . . . . . . . . 14 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
142	7, 141	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 Or 𝐴)
143	142	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑅 Or 𝐴)
144	143	3ad2antl1 1202	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑅 Or 𝐴)
145	37	sselda 3945	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺) → 𝑤 ∈ 𝐴)
146	145	3adant3 1148	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) → 𝑤 ∈ 𝐴)
147	146	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑤 ∈ 𝐴)
148		sotrieq2 5602	. . . . . . . . . . 11 ⊢ ((𝑅 Or 𝐴 ∧ (𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴)) → (𝑤 = 𝑡 ↔ (¬ 𝑤𝑅𝑡 ∧ ¬ 𝑡𝑅𝑤)))
149	144, 147, 116, 148	syl12anc 849	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → (𝑤 = 𝑡 ↔ (¬ 𝑤𝑅𝑡 ∧ ¬ 𝑡𝑅𝑤)))
150	140, 135, 149	mpbir2and 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ (𝐹‘𝑤) = (𝐹‘𝑡)) → 𝑤 = 𝑡)
151	49, 150	syldan 602	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → 𝑤 = 𝑡)
152	151	ex 417	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺) → (((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡))
153	152	3expb 1136	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ ran 𝐺 ∧ 𝑡 ∈ ran 𝐺)) → (((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡))
154	153	ralrimivva 3214	. . . . 5 ⊢ (𝜑 → ∀𝑤 ∈ ran 𝐺∀𝑡 ∈ ran 𝐺(((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡))
155		dff13 7253	. . . . 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 594	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺–1-1→ran 𝐹)
157		riotaex 7372	. . . . . . . . . . 11 ⊢ (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V
158	157	rgenw 3089	. . . . . . . . . 10 ⊢ ∀𝑢 ∈ ran 𝐹(℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V
159	35	fnmpt 6676	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ ran 𝐹(℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V → 𝐺 Fn ran 𝐹)
160	158, 159	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn ran 𝐹)
161		dffn3 6719	. . . . . . . . 9 ⊢ (𝐺 Fn ran 𝐹 ↔ 𝐺:ran 𝐹⟶ran 𝐺)
162	160, 161	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ran 𝐹⟶ran 𝐺)
163	162	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (𝐺‘𝑢) ∈ ran 𝐺)
164	163	fvresd 6902	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ((𝐹 ↾ ran 𝐺)‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝑢)))
165		simpr 489	. . . . . . . . . . . 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 7014	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (𝐺‘𝑢) = (℩𝑣 ∈ (◡𝐹 “ {𝑢})∀𝑡 ∈ (◡𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
168	167, 17	eqeltrd 2869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢}))
169		fvex 6895	. . . . . . . . . . . 12 ⊢ (𝐺‘𝑢) ∈ V
170		eleq1 2857	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐺‘𝑢) → (𝑤 ∈ (◡𝐹 “ {𝑢}) ↔ (𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢})))
171		eleq1 2857	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐺‘𝑢) → (𝑤 ∈ 𝐴 ↔ (𝐺‘𝑢) ∈ 𝐴))
172		fveqeq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐺‘𝑢) → ((𝐹‘𝑤) = 𝑢 ↔ (𝐹‘(𝐺‘𝑢)) = 𝑢))
173	171, 172	anbi12d 643	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐺‘𝑢) → ((𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢) ↔ ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢)))
174	170, 173	bibi12d 348	. . . . . . . . . . . . 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 343	. . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑤 ∈ (◡𝐹 “ {𝑢}) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) = 𝑢)))
177	169, 175, 176	vtocl 3534	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢}) ↔ ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢)))
178	177	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ((𝐺‘𝑢) ∈ (◡𝐹 “ {𝑢}) ↔ ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢)))
179	168, 178	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ((𝐺‘𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑢)) = 𝑢))
180	179	simprd 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → (𝐹‘(𝐺‘𝑢)) = 𝑢)
181	164, 180	eqtr2d 2805	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 = ((𝐹 ↾ ran 𝐺)‘(𝐺‘𝑢)))
182		fveq2 6882	. . . . . . . 8 ⊢ (𝑤 = (𝐺‘𝑢) → ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘(𝐺‘𝑢)))
183	182	rspceeqv 3613	. . . . . . 7 ⊢ (((𝐺‘𝑢) ∈ ran 𝐺 ∧ 𝑢 = ((𝐹 ↾ ran 𝐺)‘(𝐺‘𝑢))) → ∃𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤))
184	163, 181, 183	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ran 𝐹) → ∃𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤))
185	184	ralrimiva 3163	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ ran 𝐹∃𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤))
186		dffo3 7098	. . . . 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 594	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺–onto→ran 𝐹)
188		df-f1o 6544	. . . 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 594	. . 3 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺–1-1-onto→ran 𝐹)
190		reseq2 5974	. . . . 5 ⊢ (𝑣 = ran 𝐺 → (𝐹 ↾ 𝑣) = (𝐹 ↾ ran 𝐺))
191		id 23	. . . . 5 ⊢ (𝑣 = ran 𝐺 → 𝑣 = ran 𝐺)
192		eqidd 2770	. . . . 5 ⊢ (𝑣 = ran 𝐺 → ran 𝐹 = ran 𝐹)
193	190, 191, 192	f1oeq123d 6815	. . . 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 3590	. . 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 595	. 2 ⊢ (𝜑 → ∃𝑣 ∈ 𝒫 𝐴(𝐹 ↾ 𝑣):𝑣–1-1-onto→ran 𝐹)
196		reseq2 5974	. . . 4 ⊢ (𝑣 = 𝑥 → (𝐹 ↾ 𝑣) = (𝐹 ↾ 𝑥))
197		id 23	. . . 4 ⊢ (𝑣 = 𝑥 → 𝑣 = 𝑥)
198		eqidd 2770	. . . 4 ⊢ (𝑣 = 𝑥 → ran 𝐹 = ran 𝐹)
199	196, 197, 198	f1oeq123d 6815	. . 3 ⊢ (𝑣 = 𝑥 → ((𝐹 ↾ 𝑣):𝑣–1-1-onto→ran 𝐹 ↔ (𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹))
200	199	cbvrexvw 3250	. 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 221	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  ∃!wreu 3374  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594   class class class wbr 5113   ↦ cmpt 5196   Or wor 5569   We wwe 5614  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   ↾ cres 5664   “ cima 5665   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  ℩crio 7367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368

This theorem is referenced by:  wessf1orn  45795