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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 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))

Proof of Theorem f1ocan1fv
StepHypRef Expression
1 f1of 6356 . . . 4 (𝐺:𝐴1-1-onto𝐵𝐺:𝐴𝐵)
213ad2ant2 1165 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝐺:𝐴𝐵)
3 f1ocnv 6368 . . . . . 6 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵1-1-onto𝐴)
4 f1of 6356 . . . . . 6 (𝐺:𝐵1-1-onto𝐴𝐺:𝐵𝐴)
53, 4syl 17 . . . . 5 (𝐺:𝐴1-1-onto𝐵𝐺:𝐵𝐴)
653ad2ant2 1165 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝐺:𝐵𝐴)
7 simp3 1169 . . . 4 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → 𝑋𝐵)
86, 7ffvelrnd 6586 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺𝑋) ∈ 𝐴)
9 fvco3 6500 . . 3 ((𝐺:𝐴𝐵 ∧ (𝐺𝑋) ∈ 𝐴) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹‘(𝐺‘(𝐺𝑋))))
102, 8, 9syl2anc 580 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹‘(𝐺‘(𝐺𝑋))))
11 f1ocnvfv2 6761 . . . 4 ((𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺‘(𝐺𝑋)) = 𝑋)
12113adant1 1161 . . 3 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐺‘(𝐺𝑋)) = 𝑋)
1312fveq2d 6415 . 2 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → (𝐹‘(𝐺‘(𝐺𝑋))) = (𝐹𝑋))
1410, 13eqtrd 2833 1 ((Fun 𝐹𝐺:𝐴1-1-onto𝐵𝑋𝐵) → ((𝐹𝐺)‘(𝐺𝑋)) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1108   = wceq 1653  wcel 2157  ccnv 5311  ccom 5316  Fun wfun 6095  wf 6097  1-1-ontowf1o 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
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