Theorem f1ocan2fv 38226

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 6809	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → Rel 𝐺)
2		dfrel2 6175	. . . . . 6 ⊢ (Rel 𝐺 ↔ ◡◡𝐺 = 𝐺)
3	1, 2	sylib 220	. . . . 5 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡◡𝐺 = 𝐺)
4	3	3ad2ant2 1147	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ◡◡𝐺 = 𝐺)
5	4	fveq1d 6869	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → (◡◡𝐺‘𝑋) = (𝐺‘𝑋))
6	5	fveq2d 6871	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋)))
7		f1ocnv 6819	. . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐴)
8		f1ocan1fv 38225	. . 3 ⊢ ((Fun 𝐹 ∧ ◡𝐺:𝐵–1-1-onto→𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋))
9	7, 8	syl3an2 1177	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋))
10	6, 9	eqtr3d 2799	1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋)) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ^◡ccnv 5646   ∘ ccom 5651  Rel wrel 5652  Fun wfun 6515  –1-1-onto→wf1o 6520  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529

This theorem is referenced by: (None)