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Theorem domssex2 9094

Description: A corollary of disjenex 9092. If 𝐹 is an injection from 𝐴 to 𝐵 then there is a right inverse 𝑔 of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)

**Assertion**
Ref	Expression
domssex2	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑔(𝑔:𝐵–1-1→V ∧ (𝑔 ∘ 𝐹) = ( I ↾ 𝐴)))

Distinct variable groups: 𝐴,𝑔 𝐵,𝑔 𝑔,𝐹 Allowed substitution hints: 𝑉(𝑔) 𝑊(𝑔)

**Proof of Theorem domssex2**
Step	Hyp	Ref	Expression
1		f1f 6745	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fex2 7902	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
3	1, 2	syl3an1 1172	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
4		f1stres 7979	. . . . 5 ⊢ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝐹)
5		difexg 5275	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
6	5	3ad2ant3 1144	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
7		snex 5386	. . . . . 6 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
8		xpexg 7718	. . . . . 6 ⊢ (((𝐵 ∖ ran 𝐹) ∈ V ∧ {𝒫 ∪ ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ∈ V)
9	6, 7, 8	sylancl 594	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ∈ V)
10		fex2 7902	. . . . 5 ⊢ (((1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝐹) ∧ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ∈ V ∧ (𝐵 ∖ ran 𝐹) ∈ V) → (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ∈ V)
11	4, 9, 6, 10	mp3an2i 1477	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ∈ V)
12		unexg 7711	. . . 4 ⊢ ((𝐹 ∈ V ∧ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ∈ V) → (𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∈ V)
13	3, 11, 12	syl2anc 592	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∈ V)
14		cnvexg 7890	. . 3 ⊢ ((𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∈ V → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∈ V)
15	13, 14	syl 17	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∈ V)
16		eqid 2752	. . . . . . 7 ⊢ ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
17	16	domss2 9093	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∧ 𝐴 ⊆ ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∧ (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴)))
18	17	simp1d 1151	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))))
19		f1of1 6790	. . . . 5 ⊢ (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1→ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))))
20	18, 19	syl 17	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1→ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))))
21		ssv 3951	. . . 4 ⊢ ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ⊆ V
22		f1ss 6752	. . . 4 ⊢ ((^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1→ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∧ ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ⊆ V) → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1→V)
23	20, 21, 22	sylancl 594	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1→V)
24	17	simp3d 1153	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴))
25	23, 24	jca 518	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1→V ∧ (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴)))
26		f1eq1 6740	. . 3 ⊢ (𝑔 = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) → (𝑔:𝐵–1-1→V ↔ ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1→V))
27		coeq1 5818	. . . 4 ⊢ (𝑔 = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) → (𝑔 ∘ 𝐹) = (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∘ 𝐹))
28	27	eqeq1d 2754	. . 3 ⊢ (𝑔 = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) → ((𝑔 ∘ 𝐹) = ( I ↾ 𝐴) ↔ (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴)))
29	26, 28	anbi12d 640	. 2 ⊢ (𝑔 = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) → ((𝑔:𝐵–1-1→V ∧ (𝑔 ∘ 𝐹) = ( I ↾ 𝐴)) ↔ (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1→V ∧ (^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴))))
30	15, 25, 29	spcedv 3548	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑔(𝑔:𝐵–1-1→V ∧ (𝑔 ∘ 𝐹) = ( I ↾ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1095 = wceq 1550 ∃wex 1789 ∈ wcel 2132 Vcvv 3444 ∖ cdif 3892 ∪ cun 3893 ⊆ wss 3895 𝒫 cpw 4545 {csn 4572 ∪ cuni 4855 I cid 5530 × cxp 5634 ^◡ccnv 5635 ran crn 5637 ↾ cres 5638 ∘ ccom 5640 ⟶wf 6502 –1-1→wf1 6503 –1-1-onto→wf1o 6505 1^st c1st 7953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-ral 3067 df-rex 3077 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-1st 7955 df-2nd 7956 df-en 8913

This theorem is referenced by: (None)

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