Theorem foresf1o 33100

Description: From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)

Assertion

Ref	Expression
foresf1o	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem foresf1o

Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		focdmex 7968	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))
2	1	imp 412	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
3		foelrn 7107	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧))
4		fofn 6798	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
5		eqcom 2768	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑧))
6		fniniseg 7059	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑧 ∈ (◡𝐹 “ {𝑦}) ↔ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) = 𝑦)))
7	6	biimpar 483	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) = 𝑦)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
8	7	anassrs 473	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑧) = 𝑦) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
9	5, 8	sylan2br 607	. . . . . . . . . 10 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑧)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
10	4, 9	sylanl1 693	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑧)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
11	10	ex 418	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑦 = (𝐹‘𝑧) → 𝑧 ∈ (◡𝐹 “ {𝑦})))
12	11	reximdva 3176	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦})))
13	12	adantr 486	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦})))
14	3, 13	mpd 16	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
15	14	adantll 727	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
16	15	ralrimiva 3155	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
17		eleq1 2849	. . . 4 ⊢ (𝑧 = (𝑔‘𝑦) → (𝑧 ∈ (◡𝐹 “ {𝑦}) ↔ (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
18	17	ac6sg 10566	. . 3 ⊢ (𝐵 ∈ V → (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}) → ∃𝑔(𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))))
19	2, 16, 18	sylc 66	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑔(𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
20		frn 6717	. . . . 5 ⊢ (𝑔:𝐵⟶𝐴 → ran 𝑔 ⊆ 𝐴)
21	20	ad2antrl 741	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ran 𝑔 ⊆ 𝐴)
22		vex 3455	. . . . . 6 ⊢ 𝑔 ∈ V
23	22	rnex 7922	. . . . 5 ⊢ ran 𝑔 ∈ V
24	23	elpw 4561	. . . 4 ⊢ (ran 𝑔 ∈ 𝒫 𝐴 ↔ ran 𝑔 ⊆ 𝐴)
25	21, 24	sylibr 237	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ran 𝑔 ∈ 𝒫 𝐴)
26		fof 6796	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
27	26	ad2antlr 740	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝐹:𝐴⟶𝐵)
28	27, 21	fssresd 6749	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔⟶𝐵)
29		ffn 6709	. . . . . 6 ⊢ (𝑔:𝐵⟶𝐴 → 𝑔 Fn 𝐵)
30	29	ad2antrl 741	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝑔 Fn 𝐵)
31		dffn3 6722	. . . . 5 ⊢ (𝑔 Fn 𝐵 ↔ 𝑔:𝐵⟶ran 𝑔)
32	30, 31	sylib 221	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝑔:𝐵⟶ran 𝑔)
33		fvres 6904	. . . . . . . 8 ⊢ (𝑧 ∈ ran 𝑔 → ((𝐹 ↾ ran 𝑔)‘𝑧) = (𝐹‘𝑧))
34	33	adantl 487	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ((𝐹 ↾ ran 𝑔)‘𝑧) = (𝐹‘𝑧))
35	34	fveq2d 6889	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = (𝑔‘(𝐹‘𝑧)))
36		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵)
37		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑔:𝐵⟶𝐴
38		nfra1 3287	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})
39	37, 38	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
40	36, 39	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑦((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
41		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ ran 𝑔
42	40, 41	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑦(((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔)
43		simpr 490	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → (𝑔‘𝑦) = 𝑧)
44	43	fveq2d 6889	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → (𝐹‘(𝑔‘𝑦)) = (𝐹‘𝑧))
45	4	ad5antlr 748	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → 𝐹 Fn 𝐴)
46		simplrr 790	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
47	46	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
48		simplr 781	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → 𝑦 ∈ 𝐵)
49		rspa 3252	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
50	47, 48, 49	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
51		fniniseg 7059	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → ((𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}) ↔ ((𝑔‘𝑦) ∈ 𝐴 ∧ (𝐹‘(𝑔‘𝑦)) = 𝑦)))
52	51	simplbda 505	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})) → (𝐹‘(𝑔‘𝑦)) = 𝑦)
53	45, 50, 52	syl2anc 596	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → (𝐹‘(𝑔‘𝑦)) = 𝑦)
54	44, 53	eqtr3d 2798	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → (𝐹‘𝑧) = 𝑦)
55	54	fveq2d 6889	. . . . . . . 8 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → (𝑔‘(𝐹‘𝑧)) = (𝑔‘𝑦))
56	55, 43	eqtrd 2796	. . . . . . 7 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐵) ∧ (𝑔‘𝑦) = 𝑧) → (𝑔‘(𝐹‘𝑧)) = 𝑧)
57		fvelrnb 6945	. . . . . . . . 9 ⊢ (𝑔 Fn 𝐵 → (𝑧 ∈ ran 𝑔 ↔ ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = 𝑧))
58	57	biimpa 482	. . . . . . . 8 ⊢ ((𝑔 Fn 𝐵 ∧ 𝑧 ∈ ran 𝑔) → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = 𝑧)
59	30, 58	sylan 592	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = 𝑧)
60	42, 56, 59	r19.29af 3272	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘(𝐹‘𝑧)) = 𝑧)
61	35, 60	eqtrd 2796	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = 𝑧)
62	61	ralrimiva 3155	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ∀𝑧 ∈ ran 𝑔(𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = 𝑧)
63	32	ffvelcdmda 7084	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) ∈ ran 𝑔)
64		fvres 6904	. . . . . . . 8 ⊢ ((𝑔‘𝑦) ∈ ran 𝑔 → ((𝐹 ↾ ran 𝑔)‘(𝑔‘𝑦)) = (𝐹‘(𝑔‘𝑦)))
65	63, 64	syl 18	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ↾ ran 𝑔)‘(𝑔‘𝑦)) = (𝐹‘(𝑔‘𝑦)))
66	4	ad3antlr 744	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → 𝐹 Fn 𝐴)
67		simplrr 790	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
68		simpr 490	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
69	67, 68, 49	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
70	66, 69, 52	syl2anc 596	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑔‘𝑦)) = 𝑦)
71	65, 70	eqtrd 2796	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ↾ ran 𝑔)‘(𝑔‘𝑦)) = 𝑦)
72	71	ex 418	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → (𝑦 ∈ 𝐵 → ((𝐹 ↾ ran 𝑔)‘(𝑔‘𝑦)) = 𝑦))
73	40, 72	ralrimi 3261	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ∀𝑦 ∈ 𝐵 ((𝐹 ↾ ran 𝑔)‘(𝑔‘𝑦)) = 𝑦)
74	28, 32, 62, 73	2fvidf1od 7306	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔–1-1-onto→𝐵)
75		reseq2 5965	. . . . 5 ⊢ (𝑥 = ran 𝑔 → (𝐹 ↾ 𝑥) = (𝐹 ↾ ran 𝑔))
76		id 23	. . . . 5 ⊢ (𝑥 = ran 𝑔 → 𝑥 = ran 𝑔)
77		eqidd 2762	. . . . 5 ⊢ (𝑥 = ran 𝑔 → 𝐵 = 𝐵)
78	75, 76, 77	f1oeq123d 6818	. . . 4 ⊢ (𝑥 = ran 𝑔 → ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵 ↔ (𝐹 ↾ ran 𝑔):ran 𝑔–1-1-onto→𝐵))
79	78	rspcev 3577	. . 3 ⊢ ((ran 𝑔 ∈ 𝒫 𝐴 ∧ (𝐹 ↾ ran 𝑔):ran 𝑔–1-1-onto→𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵)
80	25, 74, 79	syl2anc 596	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵)
81	19, 80	exlimddv 1968	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-reg 9586  ax-inf2 9642  ax-ac2 10541

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-en 8974  df-r1 9768  df-rank 9769  df-scott 9929  df-card 10020  df-ac 10195

This theorem is used by:  rabfodom  33101