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Theorem fvco2 5163
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 A 𝑋 A) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))

Proof of Theorem fvco2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 fnsnfv 5153 . . . . . 6 ((𝐺 Fn A 𝑋 A) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
21imaeq2d 4591 . . . . 5 ((𝐺 Fn A 𝑋 A) → (𝐹 “ {(𝐺𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
3 imaco 4749 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
42, 3syl6reqr 2069 . . . 4 ((𝐺 Fn A 𝑋 A) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺𝑋)}))
54eleq2d 2085 . . 3 ((𝐺 Fn A 𝑋 A) → (x ((𝐹𝐺) “ {𝑋}) ↔ x (𝐹 “ {(𝐺𝑋)})))
65iotabidv 4811 . 2 ((𝐺 Fn A 𝑋 A) → (℩xx ((𝐹𝐺) “ {𝑋})) = (℩xx (𝐹 “ {(𝐺𝑋)})))
7 dffv3g 5095 . . 3 (𝑋 A → ((𝐹𝐺)‘𝑋) = (℩xx ((𝐹𝐺) “ {𝑋})))
87adantl 262 . 2 ((𝐺 Fn A 𝑋 A) → ((𝐹𝐺)‘𝑋) = (℩xx ((𝐹𝐺) “ {𝑋})))
9 funfvex 5113 . . . 4 ((Fun 𝐺 𝑋 dom 𝐺) → (𝐺𝑋) V)
109funfni 4921 . . 3 ((𝐺 Fn A 𝑋 A) → (𝐺𝑋) V)
11 dffv3g 5095 . . 3 ((𝐺𝑋) V → (𝐹‘(𝐺𝑋)) = (℩xx (𝐹 “ {(𝐺𝑋)})))
1210, 11syl 14 . 2 ((𝐺 Fn A 𝑋 A) → (𝐹‘(𝐺𝑋)) = (℩xx (𝐹 “ {(𝐺𝑋)})))
136, 8, 123eqtr4d 2060 1 ((𝐺 Fn A 𝑋 A) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226   wcel 1370  Vcvv 2531  {csn 3346  cima 4271  ccom 4272  cio 4788   Fn wfn 4820  cfv 4825
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833
This theorem is referenced by:  fvco  5164  fvco3  5165  ofco  5648
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