Theorem fcof1o 5940

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

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		fcof1 5934	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵)
2	1	ad2ant2rl 511	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴–1-1→𝐵)
3		fcofo 5935	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵)
4	3	3expa 1230	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵)
5	4	adantrr 479	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴–onto→𝐵)
6		df-f1o 5340	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
7	2, 5, 6	sylanbrc 417	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → 𝐹:𝐴–1-1-onto→𝐵)
8		simprl 531	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))
9	8	coeq2d 4898	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵)))
10		coass 5262	. . . 4 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺))
11		f1ococnv1 5621	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
12	7, 11	syl 14	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
13	12	coeq1d 4897	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
14		fcoi2 5526	. . . . . 6 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
15	14	ad2antlr 489	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
16	13, 15	eqtrd 2264	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺)
17	10, 16	eqtr3id 2278	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺)
18		f1ocnv 5605	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
19		f1of 5592	. . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴)
20		fcoi1 5525	. . . 4 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
21	7, 18, 19, 20	4syl 18	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
22	9, 17, 21	3eqtr3rd 2273	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → ◡𝐹 = 𝐺)
23	7, 22	jca 306	1 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = 𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   I cid 4391  ^◡ccnv 4730   ↾ cres 4733   ∘ ccom 4735  ⟶wf 5329  –1-1→wf1 5330  –onto→wfo 5331  –1-1-onto→wf1o 5332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341

This theorem is referenced by:  txswaphmeo  15115