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 8873

Description: A corollary of disjenex 8871. 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 6654	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fex2 7754	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
3	1, 2	syl3an1 1161	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
4		f1stres 7828	. . . . 5 ⊢ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})⟶(𝐵 ∖ ran 𝐹)
5		difexg 5246	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
6	5	3ad2ant3 1133	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
7		snex 5349	. . . . . 6 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
8		xpexg 7578	. . . . . 6 ⊢ (((𝐵 ∖ ran 𝐹) ∈ V ∧ {𝒫 ∪ ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ∈ V)
9	6, 7, 8	sylancl 585	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ∈ V)
10		fex2 7754	. . . . 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 1464	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ∈ V)
12		unexg 7577	. . . 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 583	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ∈ V)
14		cnvexg 7745	. . 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 2738	. . . . . . 7 ⊢ ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
17	16	domss2 8872	. . . . . 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 1140	. . . . 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 6699	. . . . 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 3941	. . . 4 ⊢ ran ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ⊆ V
22		f1ss 6660	. . . 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 585	. . 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 1142	. . 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 511	. 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 6649	. . 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 5755	. . . 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 2740	. . 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 630	. 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 3527	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 395 ∧ w3a 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 Vcvv 3422 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 𝒫 cpw 4530 {csn 4558 ∪ cuni 4836 I cid 5479 × cxp 5578 ^◡ccnv 5579 ran crn 5581 ↾ cres 5582 ∘ ccom 5584 ⟶wf 6414 –1-1→wf1 6415 –1-1-onto→wf1o 6417 1^st c1st 7802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-1st 7804 df-2nd 7805 df-en 8692

This theorem is referenced by: (None)

W3C validator