Theorem cocnvf1o 33388

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⟶1-1onto𝐴)

Assertion

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

Proof of Theorem cocnvf1o

Step	Hyp	Ref	Expression
1		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → 𝐹 = (𝐺 ∘ 𝐻))
2	1	coeq1d 5844	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐹 ∘ ◡𝐻) = ((𝐺 ∘ 𝐻) ∘ ◡𝐻))
3		coass 6281	. . . 4 ⊢ ((𝐺 ∘ 𝐻) ∘ ◡𝐻) = (𝐺 ∘ (𝐻 ∘ ◡𝐻))
4	2, 3	eqtrdi 2814	. . 3 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐹 ∘ ◡𝐻) = (𝐺 ∘ (𝐻 ∘ ◡𝐻)))
5		cocnvf1o.3	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐴⟶1-1onto𝐴)
6		f1ococnv2 6872	. . . . . . 7 ⊢ (𝐻:𝐴⟶1-1onto𝐴 → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
7	5, 6	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
8	7	coeq2d 5845	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = (𝐺 ∘ ( I ↾ 𝐴)))
9		cocnvf1o.2	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
10		fcoi1 6775	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (𝐺 ∘ ( I ↾ 𝐴)) = 𝐺)
11	9, 10	syl 18	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ ( I ↾ 𝐴)) = 𝐺)
12	8, 11	eqtrd 2798	. . . 4 ⊢ (𝜑 → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = 𝐺)
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = 𝐺)
14	4, 13	eqtr2d 2799	. 2 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
15		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
16	15	coeq1d 5844	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐺 ∘ 𝐻) = ((𝐹 ∘ ◡𝐻) ∘ 𝐻))
17		coass 6281	. . . 4 ⊢ ((𝐹 ∘ ◡𝐻) ∘ 𝐻) = (𝐹 ∘ (◡𝐻 ∘ 𝐻))
18	16, 17	eqtrdi 2814	. . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐺 ∘ 𝐻) = (𝐹 ∘ (◡𝐻 ∘ 𝐻)))
19		f1ococnv1 6874	. . . . . . 7 ⊢ (𝐻:𝐴⟶1-1onto𝐴 → (◡𝐻 ∘ 𝐻) = ( I ↾ 𝐴))
20	5, 19	syl 18	. . . . . 6 ⊢ (𝜑 → (◡𝐻 ∘ 𝐻) = ( I ↾ 𝐴))
21	20	coeq2d 5845	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = (𝐹 ∘ ( I ↾ 𝐴)))
22		cocnvf1o.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
23		fcoi1 6775	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
24	22, 23	syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
25	21, 24	eqtrd 2798	. . . 4 ⊢ (𝜑 → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = 𝐹)
26	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = 𝐹)
27	18, 26	eqtr2d 2799	. 2 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → 𝐹 = (𝐺 ∘ 𝐻))
28	14, 27	impbida 813	1 ⊢ (𝜑 → (𝐹 = (𝐺 ∘ 𝐻) ↔ 𝐺 = (𝐹 ∘ ◡𝐻)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   I cid 5549  ^◡ccnv 5654   ↾ cres 5657   ∘ ccom 5659  ⟶wf 6551  ⟶1-1ontowf1o 6554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562

This theorem is used by:  mplvrpmrhm  34246