Theorem f1ocan1fv 38660

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 6824	. . . 4 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐺:𝐴⟶𝐵)
2	1	3ad2ant2 1152	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → 𝐺:𝐴⟶𝐵)
3		f1ocnv 6837	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐴)
4		f1of 6824	. . . . . 6 ⊢ (◡𝐺:𝐵–1-1-onto→𝐴 → ◡𝐺:𝐵⟶𝐴)
5	3, 4	syl 18	. . . . 5 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵⟶𝐴)
6	5	3ad2ant2 1152	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ◡𝐺:𝐵⟶𝐴)
7		simp3 1156	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
8	6, 7	ffvelcdmd 7085	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (◡𝐺‘𝑋) ∈ 𝐴)
9		fvco3 6985	. . 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–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘(𝐺‘(◡𝐺‘𝑋))))
11		f1ocnvfv2 7285	. . . 4 ⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋)
12	11	3adant1 1148	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋)
13	12	fveq2d 6889	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝐺‘(◡𝐺‘𝑋))) = (𝐹‘𝑋))
14	10, 13	eqtrd 2796	1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ^◡ccnv 5650   ∘ ccom 5655  Fun wfun 6532  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  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-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is used by:  f1ocan2fv  38661