domssex2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > domssex2		Structured version Visualization version GIF version

Theorem domssex2 9124

Description: A corollary of disjenex 9122. 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 6774	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fex2 7932	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
3	1, 2	syl3an1 1179	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
4		f1stres 8009	. . . . 5 ⊢ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝐹)
5		difexg 5299	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
6	5	3ad2ant3 1151	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
7		snex 5410	. . . . . 6 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
8		xpexg 7748	. . . . . 6 ⊢ (((𝐵 ∖ ran 𝐹) ∈ V ∧ {𝒫 ∪ ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ∈ V)
9	6, 7, 8	sylancl 597	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ∈ V)
10		fex2 7932	. . . . 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 1493	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ∈ V)
12		unexg 7741	. . . 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 595	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∈ V)
14		cnvexg 7920	. . 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 2761	. . . . . . 7 ⊢ ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
17	16	domss2 9123	. . . . . 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 1158	. . . . 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 6819	. . . . 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 3960	. . . 4 ⊢ ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ⊆ V
22		f1ss 6781	. . . 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 597	. . 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 1160	. . 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 520	. 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 6769	. . 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 5843	. . . 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 2763	. . 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 643	. 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 3556	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 400 ∧ w3a 1101 = wceq 1568 ∃wex 1807 ∈ wcel 2141 Vcvv 3453 ∖ cdif 3901 ∪ cun 3902 ⊆ wss 3904 𝒫 cpw 4561 {csn 4588 ∪ cuni 4871 I cid 5555 × cxp 5659 ^◡ccnv 5660 ran crn 5662 ↾ cres 5663 ∘ ccom 5665 ⟶wf 6532 –1-1→wf1 6533 –1-1-onto→wf1o 6535 1^st c1st 7983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-1st 7985 df-2nd 7986 df-en 8943

This theorem is referenced by: (None)

W3C validator