Theorem f1ocan1fv 34009

Description: Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
f1ocan1fv	⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘𝑋))

Proof of Theorem f1ocan1fv

Step	Hyp	Ref	Expression
1		f1of 6356	. . . 4 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐺:𝐴⟶𝐵)
2	1	3ad2ant2 1165	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → 𝐺:𝐴⟶𝐵)
3		f1ocnv 6368	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐴)
4		f1of 6356	. . . . . 6 ⊢ (◡𝐺:𝐵–1-1-onto→𝐴 → ◡𝐺:𝐵⟶𝐴)
5	3, 4	syl 17	. . . . 5 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵⟶𝐴)
6	5	3ad2ant2 1165	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ◡𝐺:𝐵⟶𝐴)
7		simp3 1169	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
8	6, 7	ffvelrnd 6586	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (◡𝐺‘𝑋) ∈ 𝐴)
9		fvco3 6500	. . 3 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (◡𝐺‘𝑋) ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘(𝐺‘(◡𝐺‘𝑋))))
10	2, 8, 9	syl2anc 580	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘(𝐺‘(◡𝐺‘𝑋))))
11		f1ocnvfv2 6761	. . . 4 ⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋)
12	11	3adant1 1161	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋)
13	12	fveq2d 6415	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝐺‘(◡𝐺‘𝑋))) = (𝐹‘𝑋))
14	10, 13	eqtrd 2833	1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  ^◡ccnv 5311   ∘ ccom 5316  Fun wfun 6095  ⟶wf 6097  –1-1-onto→wf1o 6100  ‘cfv 6101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109

This theorem is referenced by:  f1ocan2fv  34010