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Theorem fvco2 5586
Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)
Assertion
Ref Expression
fvco2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))

Proof of Theorem fvco2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 imaco 5135 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
2 fnsnfv 5576 . . . . . 6 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
32imaeq2d 4971 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐺𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
41, 3eqtr4id 2229 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺𝑋)}))
54eleq2d 2247 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝑥 ∈ ((𝐹𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
65iotabidv 5200 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
7 dffv3g 5512 . . 3 (𝑋𝐴 → ((𝐹𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
87adantl 277 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
9 funfvex 5533 . . . 4 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝐺𝑋) ∈ V)
109funfni 5317 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) ∈ V)
11 dffv3g 5512 . . 3 ((𝐺𝑋) ∈ V → (𝐹‘(𝐺𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
1210, 11syl 14 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹‘(𝐺𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
136, 8, 123eqtr4d 2220 1 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2738  {csn 3593  cima 4630  ccom 4631  cio 5177   Fn wfn 5212  cfv 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225
This theorem is referenced by:  fvco  5587  fvco3  5588  ofco  6101  updjudhcoinlf  7079  updjudhcoinrg  7080  updjud  7081  caseinl  7090  caseinr  7091  ctm  7108  enomnilem  7136  enmkvlem  7159  enwomnilem  7167  ringidvalg  13144
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