Theorem f1ocan2fv 38801

Description: Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
f1ocan2fv	⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋)) = (𝐹‘𝑋))

Proof of Theorem f1ocan2fv

Step	Hyp	Ref	Expression
1		f1orel 6835	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → Rel 𝐺)
2		dfrel2 6186	. . . . . 6 ⊢ (Rel 𝐺 ↔ ◡◡𝐺 = 𝐺)
3	1, 2	sylib 221	. . . . 5 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡◡𝐺 = 𝐺)
4	3	3ad2ant2 1152	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ◡◡𝐺 = 𝐺)
5	4	fveq1d 6895	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → (◡◡𝐺‘𝑋) = (𝐺‘𝑋))
6	5	fveq2d 6897	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋)))
7		f1ocnv 6845	. . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐴)
8		f1ocan1fv 38800	. . 3 ⊢ ((Fun 𝐹 ∧ ◡𝐺:𝐵–1-1-onto→𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋))
9	7, 8	syl3an2 1182	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋))
10	6, 9	eqtr3d 2800	1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋)) = (𝐹‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ^◡ccnv 5653   ∘ ccom 5658  Rel wrel 5659  Fun wfun 6540  –1-1-onto→wf1o 6545  ‘cfv 6546

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554

This theorem is used by: (None)