domssex2 - Metamath Proof Explorer

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Theorem domssex2 9157

Description: A corollary of disjenex 9155. 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 6778	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fex2 7948	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
3	1, 2	syl3an1 1181	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
4		f1stres 8025	. . . . 5 ⊢ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝐹)
5		difexg 5290	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
6	5	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
7		snex 5396	. . . . . 6 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
8		xpexg 7764	. . . . . 6 ⊢ (((𝐵 ∖ ran 𝐹) ∈ V ∧ {𝒫 ∪ ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ∈ V)
9	6, 7, 8	sylancl 598	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ∈ V)
10		fex2 7948	. . . . 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 1495	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ∈ V)
12		unexg 7760	. . . 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 596	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∈ V)
14		cnvexg 7936	. . 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 18	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∈ V)
16		eqid 2760	. . . . . . 7 ⊢ ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
17	16	domss2 9156	. . . . . 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 1160	. . . . 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 6823	. . . . 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 18	. . . 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 3932	. . . 4 ⊢ ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ⊆ V
22		f1ss 6785	. . . 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 598	. . 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 1162	. . 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 521	. 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 6773	. . 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 5834	. . . 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 2762	. . 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 644	. 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 3552	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∃wex 1811 ∈ wcel 2144 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ∪ cun 4149 𝒫 cpw 4556 {csn 4583 ∪ cuni 4866 I cid 5544 × cxp 5648 ^◡ccnv 5649 ran crn 5651 ↾ cres 5652 ∘ ccom 5654 ⟶wf 6534 –1-1→wf1 6535 –1-1-onto→wf1o 6537 1^st c1st 7999

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 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-1st 8001 df-2nd 8002 df-en 8975

This theorem is used by: (None)

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