Theorem f1ocan1fv 38928

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⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘𝑋))

Proof of Theorem f1ocan1fv

Step	Hyp	Ref	Expression
1		f1of 6854	. . . 4 ⊢ (𝐺:𝐴⟶1-1onto𝐵 → 𝐺:𝐴⟶𝐵)
2	1	3ad2ant2 1152	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → 𝐺:𝐴⟶𝐵)
3		f1ocnv 6867	. . . . . 6 ⊢ (𝐺:𝐴⟶1-1onto𝐵 → ◡𝐺:𝐵⟶1-1onto𝐴)
4		f1of 6854	. . . . . 6 ⊢ (◡𝐺:𝐵⟶1-1onto𝐴 → ◡𝐺:𝐵⟶𝐴)
5	3, 4	syl 18	. . . . 5 ⊢ (𝐺:𝐴⟶1-1onto𝐵 → ◡𝐺:𝐵⟶𝐴)
6	5	3ad2ant2 1152	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → ◡𝐺:𝐵⟶𝐴)
7		simp3 1156	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
8	6, 7	ffvelcdmd 7117	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → (◡𝐺‘𝑋) ∈ 𝐴)
9		fvco3 7016	. . 3 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (◡𝐺‘𝑋) ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘(𝐺‘(◡𝐺‘𝑋))))
10	2, 8, 9	syl2anc 596	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘(𝐺‘(◡𝐺‘𝑋))))
11		f1ocnvfv2 7316	. . . 4 ⊢ ((𝐺:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋)
12	11	3adant1 1148	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋)
13	12	fveq2d 6920	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝐺‘(◡𝐺‘𝑋))) = (𝐹‘𝑋))
14	10, 13	eqtrd 2798	1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶1-1onto𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ^◡ccnv 5657   ∘ ccom 5662  Fun wfun 6560  ⟶wf 6562  ⟶1-1ontowf1o 6565  ‘cfv 6566

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-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  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 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by:  f1ocan2fv  38929