Theorem cocnvf1o 33187

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 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → 𝐹 = (𝐺 ∘ 𝐻))
2	1	coeq1d 5845	. . . 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 6849	. . . . . . 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 5846	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = (𝐺 ∘ ( I ↾ 𝐴)))
9		cocnvf1o.2	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
10		fcoi1 6753	. . . . . 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 486	. . 3 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = 𝐺)
14	4, 13	eqtr2d 2798	. 2 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
15		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
16	15	coeq1d 5845	. . . 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 6851	. . . . . . 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 5846	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = (𝐹 ∘ ( I ↾ 𝐴)))
22		cocnvf1o.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
23		fcoi1 6753	. . . . . 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 486	. . 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 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 5553  ^◡ccnv 5658   ↾ cres 5661   ∘ ccom 5663  ⟶wf 6533  –1-1-onto→wf1o 6536

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544

This theorem is used by:  mplvrpmrhm  34044