Theorem foresf1o 29343

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		fornex 7135	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))
2	1	imp 445	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
3		foelrn 6378	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧))
4		fofn 6117	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
5		eqcom 2629	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑧))
6		fniniseg 6338	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑧 ∈ (◡𝐹 “ {𝑦}) ↔ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) = 𝑦)))
7	6	biimpar 502	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) = 𝑦)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
8	7	anassrs 680	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑧) = 𝑦) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
9	5, 8	sylan2br 493	. . . . . . . . . 10 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑧)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
10	4, 9	sylanl1 682	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑧)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
11	10	ex 450	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑦 = (𝐹‘𝑧) → 𝑧 ∈ (◡𝐹 “ {𝑦})))
12	11	reximdva 3017	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦})))
13	12	adantr 481	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦})))
14	3, 13	mpd 15	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
15	14	adantll 750	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
16	15	ralrimiva 2966	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
17		eleq1 2689	. . . 4 ⊢ (𝑧 = (𝑔‘𝑦) → (𝑧 ∈ (◡𝐹 “ {𝑦}) ↔ (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
18	17	ac6sg 9310	. . 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 65	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑔(𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
20		frn 6053	. . . . 5 ⊢ (𝑔:𝐵⟶𝐴 → ran 𝑔 ⊆ 𝐴)
21	20	ad2antrl 764	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ran 𝑔 ⊆ 𝐴)
22		vex 3203	. . . . . 6 ⊢ 𝑔 ∈ V
23	22	rnex 7100	. . . . 5 ⊢ ran 𝑔 ∈ V
24	23	elpw 4164	. . . 4 ⊢ (ran 𝑔 ∈ 𝒫 𝐴 ↔ ran 𝑔 ⊆ 𝐴)
25	21, 24	sylibr 224	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ran 𝑔 ∈ 𝒫 𝐴)
26		fof 6115	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
27	26	ad2antlr 763	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝐹:𝐴⟶𝐵)
28	27, 21	fssresd 6071	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔⟶𝐵)
29		ffn 6045	. . . . . 6 ⊢ (𝑔:𝐵⟶𝐴 → 𝑔 Fn 𝐵)
30	29	ad2antrl 764	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝑔 Fn 𝐵)
31		dffn3 6054	. . . . 5 ⊢ (𝑔 Fn 𝐵 ↔ 𝑔:𝐵⟶ran 𝑔)
32	30, 31	sylib 208	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝑔:𝐵⟶ran 𝑔)
33		fvres 6207	. . . . . . . 8 ⊢ (𝑧 ∈ ran 𝑔 → ((𝐹 ↾ ran 𝑔)‘𝑧) = (𝐹‘𝑧))
34	33	adantl 482	. . . . . . 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 6195	. . . . . 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 1843	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵)
37		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑔:𝐵⟶𝐴
38		nfra1 2941	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})
39	37, 38	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
40	36, 39	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑦((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
41		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ ran 𝑔
42	40, 41	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑦(((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔)
43		simpr 477	. . . . . . . . . . 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 6195	. . . . . . . . . 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 771	. . . . . . . . . . 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 801	. . . . . . . . . . . . 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 762	. . . . . . . . . . . 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 792	. . . . . . . . . . . 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 2930	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
50	47, 48, 49	syl2anc 693	. . . . . . . . . . 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 6338	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → ((𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}) ↔ ((𝑔‘𝑦) ∈ 𝐴 ∧ (𝐹‘(𝑔‘𝑦)) = 𝑦)))
52	51	simplbda 654	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})) → (𝐹‘(𝑔‘𝑦)) = 𝑦)
53	45, 50, 52	syl2anc 693	. . . . . . . . . 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 2658	. . . . . . . . 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 6195	. . . . . . . 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 2656	. . . . . . 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 6243	. . . . . . . . 9 ⊢ (𝑔 Fn 𝐵 → (𝑧 ∈ ran 𝑔 ↔ ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = 𝑧))
58	57	biimpa 501	. . . . . . . 8 ⊢ ((𝑔 Fn 𝐵 ∧ 𝑧 ∈ ran 𝑔) → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = 𝑧)
59	30, 58	sylan 488	. . . . . . 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 3076	. . . . . 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 2656	. . . . 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 2966	. . . 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	ffvelrnda 6359	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) ∈ ran 𝑔)
64		fvres 6207	. . . . . . . 8 ⊢ ((𝑔‘𝑦) ∈ ran 𝑔 → ((𝐹 ↾ ran 𝑔)‘(𝑔‘𝑦)) = (𝐹‘(𝑔‘𝑦)))
65	63, 64	syl 17	. . . . . . 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 767	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → 𝐹 Fn 𝐴)
67		simplrr 801	. . . . . . . . 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 477	. . . . . . . . 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 693	. . . . . . . 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 693	. . . . . . 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 2656	. . . . . 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 450	. . . . 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 2957	. . . 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 6553	. . 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 5391	. . . . 5 ⊢ (𝑥 = ran 𝑔 → (𝐹 ↾ 𝑥) = (𝐹 ↾ ran 𝑔))
76		id 22	. . . . 5 ⊢ (𝑥 = ran 𝑔 → 𝑥 = ran 𝑔)
77		eqidd 2623	. . . . 5 ⊢ (𝑥 = ran 𝑔 → 𝐵 = 𝐵)
78	75, 76, 77	f1oeq123d 6133	. . . 4 ⊢ (𝑥 = ran 𝑔 → ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵 ↔ (𝐹 ↾ ran 𝑔):ran 𝑔–1-1-onto→𝐵))
79	78	rspcev 3309	. . 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 693	. 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 1863	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-en 7956  df-r1 8627  df-rank 8628  df-card 8765  df-ac 8939

This theorem is referenced by:  rabfodom  29344