Theorem foresf1o 32523

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 7980	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))
2	1	imp 406	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
3		foelrn 7127	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧))
4		fofn 6822	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
5		eqcom 2744	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑧))
6		fniniseg 7080	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑧 ∈ (◡𝐹 “ {𝑦}) ↔ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) = 𝑦)))
7	6	biimpar 477	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) = 𝑦)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
8	7	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑧) = 𝑦) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
9	5, 8	sylan2br 595	. . . . . . . . . 10 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑧)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
10	4, 9	sylanl1 680	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑧)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
11	10	ex 412	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑦 = (𝐹‘𝑧) → 𝑧 ∈ (◡𝐹 “ {𝑦})))
12	11	reximdva 3168	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦})))
13	12	adantr 480	. . . . . 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 714	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
16	15	ralrimiva 3146	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
17		eleq1 2829	. . . 4 ⊢ (𝑧 = (𝑔‘𝑦) → (𝑧 ∈ (◡𝐹 “ {𝑦}) ↔ (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
18	17	ac6sg 10528	. . 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 6743	. . . . 5 ⊢ (𝑔:𝐵⟶𝐴 → ran 𝑔 ⊆ 𝐴)
21	20	ad2antrl 728	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ran 𝑔 ⊆ 𝐴)
22		vex 3484	. . . . . 6 ⊢ 𝑔 ∈ V
23	22	rnex 7932	. . . . 5 ⊢ ran 𝑔 ∈ V
24	23	elpw 4604	. . . 4 ⊢ (ran 𝑔 ∈ 𝒫 𝐴 ↔ ran 𝑔 ⊆ 𝐴)
25	21, 24	sylibr 234	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ran 𝑔 ∈ 𝒫 𝐴)
26		fof 6820	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
27	26	ad2antlr 727	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝐹:𝐴⟶𝐵)
28	27, 21	fssresd 6775	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔⟶𝐵)
29		ffn 6736	. . . . . 6 ⊢ (𝑔:𝐵⟶𝐴 → 𝑔 Fn 𝐵)
30	29	ad2antrl 728	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝑔 Fn 𝐵)
31		dffn3 6748	. . . . 5 ⊢ (𝑔 Fn 𝐵 ↔ 𝑔:𝐵⟶ran 𝑔)
32	30, 31	sylib 218	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝑔:𝐵⟶ran 𝑔)
33		fvres 6925	. . . . . . . 8 ⊢ (𝑧 ∈ ran 𝑔 → ((𝐹 ↾ ran 𝑔)‘𝑧) = (𝐹‘𝑧))
34	33	adantl 481	. . . . . . 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 6910	. . . . . 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 1914	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵)
37		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑔:𝐵⟶𝐴
38		nfra1 3284	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})
39	37, 38	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
40	36, 39	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑦((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
41		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ ran 𝑔
42	40, 41	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑦(((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔)
43		simpr 484	. . . . . . . . . . 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 6910	. . . . . . . . . 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 735	. . . . . . . . . . 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 778	. . . . . . . . . . . . 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 726	. . . . . . . . . . . 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 769	. . . . . . . . . . . 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 3248	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
50	47, 48, 49	syl2anc 584	. . . . . . . . . . 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 7080	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → ((𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}) ↔ ((𝑔‘𝑦) ∈ 𝐴 ∧ (𝐹‘(𝑔‘𝑦)) = 𝑦)))
52	51	simplbda 499	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})) → (𝐹‘(𝑔‘𝑦)) = 𝑦)
53	45, 50, 52	syl2anc 584	. . . . . . . . . 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 2779	. . . . . . . . 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 6910	. . . . . . . 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 2777	. . . . . . 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 6969	. . . . . . . . 9 ⊢ (𝑔 Fn 𝐵 → (𝑧 ∈ ran 𝑔 ↔ ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = 𝑧))
58	57	biimpa 476	. . . . . . . 8 ⊢ ((𝑔 Fn 𝐵 ∧ 𝑧 ∈ ran 𝑔) → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = 𝑧)
59	30, 58	sylan 580	. . . . . . 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 3268	. . . . . 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 2777	. . . . 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 3146	. . . 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 7104	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) ∈ ran 𝑔)
64		fvres 6925	. . . . . . . 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 731	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → 𝐹 Fn 𝐴)
67		simplrr 778	. . . . . . . . 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 484	. . . . . . . . 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 584	. . . . . . . 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 584	. . . . . . 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 2777	. . . . . 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 412	. . . . 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 3257	. . . 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 7318	. . 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 5992	. . . . 5 ⊢ (𝑥 = ran 𝑔 → (𝐹 ↾ 𝑥) = (𝐹 ↾ ran 𝑔))
76		id 22	. . . . 5 ⊢ (𝑥 = ran 𝑔 → 𝑥 = ran 𝑔)
77		eqidd 2738	. . . . 5 ⊢ (𝑥 = ran 𝑔 → 𝐵 = 𝐵)
78	75, 76, 77	f1oeq123d 6842	. . . 4 ⊢ (𝑥 = ran 𝑔 → ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵 ↔ (𝐹 ↾ ran 𝑔):ran 𝑔–1-1-onto→𝐵))
79	78	rspcev 3622	. . 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 584	. 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 1935	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  𝒫 cpw 4600  {csn 4626  ^◡ccnv 5684  ran crn 5686   ↾ cres 5687   “ cima 5688   Fn wfn 6556  ⟶wf 6557  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-en 8986  df-r1 9804  df-rank 9805  df-card 9979  df-ac 10156

This theorem is referenced by:  rabfodom  32524