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Theorem fvco2 5242
 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 fnsnfv 5232 . . . . . 6 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
21imaeq2d 4668 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐺𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
3 imaco 4826 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
42, 3syl6reqr 2091 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺𝑋)}))
54eleq2d 2107 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝑥 ∈ ((𝐹𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
65iotabidv 4888 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
7 dffv3g 5174 . . 3 (𝑋𝐴 → ((𝐹𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
87adantl 262 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
9 funfvex 5192 . . . 4 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝐺𝑋) ∈ V)
109funfni 4999 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) ∈ V)
11 dffv3g 5174 . . 3 ((𝐺𝑋) ∈ V → (𝐹‘(𝐺𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
1210, 11syl 14 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹‘(𝐺𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
136, 8, 123eqtr4d 2082 1 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375   “ cima 4348   ∘ ccom 4349  ℩cio 4865   Fn wfn 4897  ‘cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910 This theorem is referenced by:  fvco  5243  fvco3  5244  ofco  5729
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