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Theorem fvco2 5185
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 5175 . . . . . 6 ((𝐺 Fn A 𝑋 A) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
21imaeq2d 4611 . . . . 5 ((𝐺 Fn A 𝑋 A) → (𝐹 “ {(𝐺𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
3 imaco 4769 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
42, 3syl6reqr 2088 . . . 4 ((𝐺 Fn A 𝑋 A) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺𝑋)}))
54eleq2d 2104 . . 3 ((𝐺 Fn A 𝑋 A) → (x ((𝐹𝐺) “ {𝑋}) ↔ x (𝐹 “ {(𝐺𝑋)})))
65iotabidv 4831 . 2 ((𝐺 Fn A 𝑋 A) → (℩xx ((𝐹𝐺) “ {𝑋})) = (℩xx (𝐹 “ {(𝐺𝑋)})))
7 dffv3g 5117 . . 3 (𝑋 A → ((𝐹𝐺)‘𝑋) = (℩xx ((𝐹𝐺) “ {𝑋})))
87adantl 262 . 2 ((𝐺 Fn A 𝑋 A) → ((𝐹𝐺)‘𝑋) = (℩xx ((𝐹𝐺) “ {𝑋})))
9 funfvex 5135 . . . 4 ((Fun 𝐺 𝑋 dom 𝐺) → (𝐺𝑋) V)
109funfni 4942 . . 3 ((𝐺 Fn A 𝑋 A) → (𝐺𝑋) V)
11 dffv3g 5117 . . 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 2079 1 ((𝐺 Fn A 𝑋 A) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  cima 4291  ccom 4292  cio 4808   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fvco  5186  fvco3  5187  ofco  5671
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