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Theorem fcof1o 7230

Description: Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by AV, 15-Dec-2019.)

**Assertion**
Ref	Expression
fcof1o	⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴–1-1-onto→𝐵 ∧ ^◡𝐹 = 𝐺))

**Proof of Theorem fcof1o**
Step	Hyp	Ref	Expression
1		simpll 766	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴⟶𝐵)
2		simplr 768	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → 𝐺:𝐵⟶𝐴)
3		simprr 772	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
4		simprl 770	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))
5	1, 2, 3, 4	fcof1od 7228	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴–1-1-onto→𝐵)
6	1, 2, 3, 4	2fcoidinvd 7229	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → ^◡𝐹 = 𝐺)
7	5, 6	jca 511	1 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴–1-1-onto→𝐵 ∧ ^◡𝐹 = 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 I cid 5508 ^◡ccnv 5613 ↾ cres 5616 ∘ ccom 5618 ⟶wf 6477 –1-1-onto→wf1o 6480

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489

This theorem is referenced by: setcinv 17997 yonedainv 18187 rngcinv 20552 ringcinv 20586 txswaphmeo 23720 rngcinvALTV 48386 ringcinvALTV 48420

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