Theorem cocnvf1o 32933

Description: Composing with the inverse of a bijection. (Contributed by Thierry Arnoux, 15-Jan-2026.)

Hypotheses

Ref	Expression
cocnvf1o.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
cocnvf1o.2	⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
cocnvf1o.3	⊢ (𝜑 → 𝐻:𝐴–1-1-onto→𝐴)

Assertion

Ref	Expression
cocnvf1o	⊢ (𝜑 → (𝐹 = (𝐺 ∘ 𝐻) ↔ 𝐺 = (𝐹 ∘ ◡𝐻)))

Proof of Theorem cocnvf1o

Step	Hyp	Ref	Expression
1		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → 𝐹 = (𝐺 ∘ 𝐻))
2	1	coeq1d 5835	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐹 ∘ ◡𝐻) = ((𝐺 ∘ 𝐻) ∘ ◡𝐻))
3		coass 6255	. . . 4 ⊢ ((𝐺 ∘ 𝐻) ∘ ◡𝐻) = (𝐺 ∘ (𝐻 ∘ ◡𝐻))
4	2, 3	eqtrdi 2815	. . 3 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐹 ∘ ◡𝐻) = (𝐺 ∘ (𝐻 ∘ ◡𝐻)))
5		cocnvf1o.3	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐴–1-1-onto→𝐴)
6		f1ococnv2 6836	. . . . . . 7 ⊢ (𝐻:𝐴–1-1-onto→𝐴 → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
7	5, 6	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
8	7	coeq2d 5836	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = (𝐺 ∘ ( I ↾ 𝐴)))
9		cocnvf1o.2	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
10		fcoi1 6740	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (𝐺 ∘ ( I ↾ 𝐴)) = 𝐺)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ ( I ↾ 𝐴)) = 𝐺)
12	8, 11	eqtrd 2799	. . . 4 ⊢ (𝜑 → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = 𝐺)
13	12	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = 𝐺)
14	4, 13	eqtr2d 2800	. 2 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
15		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
16	15	coeq1d 5835	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐺 ∘ 𝐻) = ((𝐹 ∘ ◡𝐻) ∘ 𝐻))
17		coass 6255	. . . 4 ⊢ ((𝐹 ∘ ◡𝐻) ∘ 𝐻) = (𝐹 ∘ (◡𝐻 ∘ 𝐻))
18	16, 17	eqtrdi 2815	. . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐺 ∘ 𝐻) = (𝐹 ∘ (◡𝐻 ∘ 𝐻)))
19		f1ococnv1 6838	. . . . . . 7 ⊢ (𝐻:𝐴–1-1-onto→𝐴 → (◡𝐻 ∘ 𝐻) = ( I ↾ 𝐴))
20	5, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (◡𝐻 ∘ 𝐻) = ( I ↾ 𝐴))
21	20	coeq2d 5836	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = (𝐹 ∘ ( I ↾ 𝐴)))
22		cocnvf1o.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
23		fcoi1 6740	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
24	22, 23	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
25	21, 24	eqtrd 2799	. . . 4 ⊢ (𝜑 → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = 𝐹)
26	25	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = 𝐹)
27	18, 26	eqtr2d 2800	. 2 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → 𝐹 = (𝐺 ∘ 𝐻))
28	14, 27	impbida 810	1 ⊢ (𝜑 → (𝐹 = (𝐺 ∘ 𝐻) ↔ 𝐺 = (𝐹 ∘ ◡𝐻)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   I cid 5543  ^◡ccnv 5648   ↾ cres 5651   ∘ ccom 5653  ⟶wf 6519  –1-1-onto→wf1o 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530

This theorem is referenced by:  mplvrpmrhm  33846