Theorem cocnvf1o 33085

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 489	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → 𝐹 = (𝐺 ∘ 𝐻))
2	1	coeq1d 5846	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐹 ∘ ◡𝐻) = ((𝐺 ∘ 𝐻) ∘ ◡𝐻))
3		coass 6266	. . . 4 ⊢ ((𝐺 ∘ 𝐻) ∘ ◡𝐻) = (𝐺 ∘ (𝐻 ∘ ◡𝐻))
4	2, 3	eqtrdi 2813	. . 3 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐹 ∘ ◡𝐻) = (𝐺 ∘ (𝐻 ∘ ◡𝐻)))
5		cocnvf1o.3	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐴–1-1-onto→𝐴)
6		f1ococnv2 6848	. . . . . . 7 ⊢ (𝐻:𝐴–1-1-onto→𝐴 → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
7	5, 6	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
8	7	coeq2d 5847	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = (𝐺 ∘ ( I ↾ 𝐴)))
9		cocnvf1o.2	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
10		fcoi1 6752	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (𝐺 ∘ ( I ↾ 𝐴)) = 𝐺)
11	9, 10	syl 18	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ ( I ↾ 𝐴)) = 𝐺)
12	8, 11	eqtrd 2797	. . . 4 ⊢ (𝜑 → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = 𝐺)
13	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = 𝐺)
14	4, 13	eqtr2d 2798	. 2 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
15		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
16	15	coeq1d 5846	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐺 ∘ 𝐻) = ((𝐹 ∘ ◡𝐻) ∘ 𝐻))
17		coass 6266	. . . 4 ⊢ ((𝐹 ∘ ◡𝐻) ∘ 𝐻) = (𝐹 ∘ (◡𝐻 ∘ 𝐻))
18	16, 17	eqtrdi 2813	. . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐺 ∘ 𝐻) = (𝐹 ∘ (◡𝐻 ∘ 𝐻)))
19		f1ococnv1 6850	. . . . . . 7 ⊢ (𝐻:𝐴–1-1-onto→𝐴 → (◡𝐻 ∘ 𝐻) = ( I ↾ 𝐴))
20	5, 19	syl 18	. . . . . 6 ⊢ (𝜑 → (◡𝐻 ∘ 𝐻) = ( I ↾ 𝐴))
21	20	coeq2d 5847	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = (𝐹 ∘ ( I ↾ 𝐴)))
22		cocnvf1o.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
23		fcoi1 6752	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
24	22, 23	syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
25	21, 24	eqtrd 2797	. . . 4 ⊢ (𝜑 → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = 𝐹)
26	25	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = 𝐹)
27	18, 26	eqtr2d 2798	. 2 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → 𝐹 = (𝐺 ∘ 𝐻))
28	14, 27	impbida 812	1 ⊢ (𝜑 → (𝐹 = (𝐺 ∘ 𝐻) ↔ 𝐺 = (𝐹 ∘ ◡𝐻)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   I cid 5554  ^◡ccnv 5659   ↾ cres 5662   ∘ ccom 5664  ⟶wf 6532  –1-1-onto→wf1o 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543

This theorem is used by:  mplvrpmrhm  33946