Theorem cocnvf1o 33240

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 5836	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐹 ∘ ◡𝐻) = ((𝐺 ∘ 𝐻) ∘ ◡𝐻))
3		coass 6257	. . . 4 ⊢ ((𝐺 ∘ 𝐻) ∘ ◡𝐻) = (𝐺 ∘ (𝐻 ∘ ◡𝐻))
4	2, 3	eqtrdi 2811	. . 3 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐹 ∘ ◡𝐻) = (𝐺 ∘ (𝐻 ∘ ◡𝐻)))
5		cocnvf1o.3	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐴–1-1-onto→𝐴)
6		f1ococnv2 6841	. . . . . . 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 5837	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = (𝐺 ∘ ( I ↾ 𝐴)))
9		cocnvf1o.2	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
10		fcoi1 6745	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (𝐺 ∘ ( I ↾ 𝐴)) = 𝐺)
11	9, 10	syl 18	. . . . 5 ⊢ (𝜑 → (𝐺 ∘ ( I ↾ 𝐴)) = 𝐺)
12	8, 11	eqtrd 2795	. . . 4 ⊢ (𝜑 → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = 𝐺)
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → (𝐺 ∘ (𝐻 ∘ ◡𝐻)) = 𝐺)
14	4, 13	eqtr2d 2796	. 2 ⊢ ((𝜑 ∧ 𝐹 = (𝐺 ∘ 𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
15		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → 𝐺 = (𝐹 ∘ ◡𝐻))
16	15	coeq1d 5836	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐺 ∘ 𝐻) = ((𝐹 ∘ ◡𝐻) ∘ 𝐻))
17		coass 6257	. . . 4 ⊢ ((𝐹 ∘ ◡𝐻) ∘ 𝐻) = (𝐹 ∘ (◡𝐻 ∘ 𝐻))
18	16, 17	eqtrdi 2811	. . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐺 ∘ 𝐻) = (𝐹 ∘ (◡𝐻 ∘ 𝐻)))
19		f1ococnv1 6843	. . . . . . 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 5837	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = (𝐹 ∘ ( I ↾ 𝐴)))
22		cocnvf1o.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
23		fcoi1 6745	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
24	22, 23	syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
25	21, 24	eqtrd 2795	. . . 4 ⊢ (𝜑 → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = 𝐹)
26	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐺 = (𝐹 ∘ ◡𝐻)) → (𝐹 ∘ (◡𝐻 ∘ 𝐻)) = 𝐹)
27	18, 26	eqtr2d 2796	. 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 5542  ^◡ccnv 5647   ↾ cres 5650   ∘ ccom 5652  ⟶wf 6524  –1-1-onto→wf1o 6527

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 2213  ax-ext 2732  ax-sep 5249  ax-pr 5391

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535

This theorem is used by:  mplvrpmrhm  34098