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Theorem fvco2 5585
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 5134 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
2 fnsnfv 5575 . . . . . 6 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
32imaeq2d 4970 . . . . 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 5199 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
7 dffv3g 5511 . . 3 (𝑋𝐴 → ((𝐹𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
87adantl 277 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
9 funfvex 5532 . . . 4 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝐺𝑋) ∈ V)
109funfni 5316 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) ∈ V)
11 dffv3g 5511 . . 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 2737  {csn 3592  cima 4629  ccom 4630  cio 5176   Fn wfn 5211  cfv 5216
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 4121  ax-pow 4174  ax-pr 4209
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 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224
This theorem is referenced by:  fvco  5586  fvco3  5587  ofco  6100  updjudhcoinlf  7078  updjudhcoinrg  7079  updjud  7080  caseinl  7089  caseinr  7090  ctm  7107  enomnilem  7135  enmkvlem  7158  enwomnilem  7166  ringidvalg  13097
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