Theorem foresf1o 31431

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 7888	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))
2	1	imp 407	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V)
3		foelrn 7056	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧))
4		fofn 6758	. . . . . . . . . 10 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
5		eqcom 2743	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑧))
6		fniniseg 7010	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑧 ∈ (◡𝐹 “ {𝑦}) ↔ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) = 𝑦)))
7	6	biimpar 478	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) = 𝑦)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
8	7	anassrs 468	. . . . . . . . . . 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 678	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑧)) → 𝑧 ∈ (◡𝐹 “ {𝑦}))
11	10	ex 413	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑦 = (𝐹‘𝑧) → 𝑧 ∈ (◡𝐹 “ {𝑦})))
12	11	reximdva 3165	. . . . . . 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 712	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
16	15	ralrimiva 3143	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑧 ∈ (◡𝐹 “ {𝑦}))
17		eleq1 2825	. . . 4 ⊢ (𝑧 = (𝑔‘𝑦) → (𝑧 ∈ (◡𝐹 “ {𝑦}) ↔ (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
18	17	ac6sg 10424	. . 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 6675	. . . . 5 ⊢ (𝑔:𝐵⟶𝐴 → ran 𝑔 ⊆ 𝐴)
21	20	ad2antrl 726	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ran 𝑔 ⊆ 𝐴)
22		vex 3449	. . . . . 6 ⊢ 𝑔 ∈ V
23	22	rnex 7849	. . . . 5 ⊢ ran 𝑔 ∈ V
24	23	elpw 4564	. . . 4 ⊢ (ran 𝑔 ∈ 𝒫 𝐴 ↔ ran 𝑔 ⊆ 𝐴)
25	21, 24	sylibr 233	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → ran 𝑔 ∈ 𝒫 𝐴)
26		fof 6756	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
27	26	ad2antlr 725	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝐹:𝐴⟶𝐵)
28	27, 21	fssresd 6709	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔⟶𝐵)
29		ffn 6668	. . . . . 6 ⊢ (𝑔:𝐵⟶𝐴 → 𝑔 Fn 𝐵)
30	29	ad2antrl 726	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝑔 Fn 𝐵)
31		dffn3 6681	. . . . 5 ⊢ (𝑔 Fn 𝐵 ↔ 𝑔:𝐵⟶ran 𝑔)
32	30, 31	sylib 217	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) → 𝑔:𝐵⟶ran 𝑔)
33		fvres 6861	. . . . . . . 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 6846	. . . . . 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 1917	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵)
37		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑔:𝐵⟶𝐴
38		nfra1 3267	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})
39	37, 38	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))
40	36, 39	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑦((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦})))
41		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ ran 𝑔
42	40, 41	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑦(((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔)
43		simpr 485	. . . . . . . . . . 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 6846	. . . . . . . . . 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 733	. . . . . . . . . . 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 776	. . . . . . . . . . . . 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 724	. . . . . . . . . . . 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 767	. . . . . . . . . . . 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 3231	. . . . . . . . . . . 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 7010	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → ((𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}) ↔ ((𝑔‘𝑦) ∈ 𝐴 ∧ (𝐹‘(𝑔‘𝑦)) = 𝑦)))
52	51	simplbda 500	. . . . . . . . . . 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 2778	. . . . . . . . 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 6846	. . . . . . . 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 2776	. . . . . . 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 6903	. . . . . . . . 9 ⊢ (𝑔 Fn 𝐵 → (𝑧 ∈ ran 𝑔 ↔ ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = 𝑧))
58	57	biimpa 477	. . . . . . . 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 3251	. . . . . 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 2776	. . . . 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 3143	. . . 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 7035	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) ∈ ran 𝑔)
64		fvres 6861	. . . . . . . 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 729	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑔:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) ∈ (◡𝐹 “ {𝑦}))) ∧ 𝑦 ∈ 𝐵) → 𝐹 Fn 𝐴)
67		simplrr 776	. . . . . . . . 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 485	. . . . . . . . 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 2776	. . . . . 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 413	. . . . 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 3240	. . . 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 7244	. . 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 5932	. . . . 5 ⊢ (𝑥 = ran 𝑔 → (𝐹 ↾ 𝑥) = (𝐹 ↾ ran 𝑔))
76		id 22	. . . . 5 ⊢ (𝑥 = ran 𝑔 → 𝑥 = ran 𝑔)
77		eqidd 2737	. . . . 5 ⊢ (𝑥 = ran 𝑔 → 𝐵 = 𝐵)
78	75, 76, 77	f1oeq123d 6778	. . . 4 ⊢ (𝑥 = ran 𝑔 → ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵 ↔ (𝐹 ↾ ran 𝑔):ran 𝑔–1-1-onto→𝐵))
79	78	rspcev 3581	. . 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 1938	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ⊆ wss 3910  𝒫 cpw 4560  {csn 4586  ^◡ccnv 5632  ran crn 5634   ↾ cres 5635   “ cima 5636   Fn wfn 6491  ⟶wf 6492  –onto→wfo 6494  –1-1-onto→wf1o 6495  ‘cfv 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577  ax-ac2 10399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-en 8884  df-r1 9700  df-rank 9701  df-card 9875  df-ac 10052

This theorem is referenced by:  rabfodom  31432