ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvco2 GIF version

Theorem fvco2 5483
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 5473 . . . . . 6 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
21imaeq2d 4876 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐺𝑋)}) = (𝐹 “ (𝐺 “ {𝑋})))
3 imaco 5039 . . . . 5 ((𝐹𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋}))
42, 3syl6reqr 2189 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {(𝐺𝑋)}))
54eleq2d 2207 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝑥 ∈ ((𝐹𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
65iotabidv 5104 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
7 dffv3g 5410 . . 3 (𝑋𝐴 → ((𝐹𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
87adantl 275 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹𝐺) “ {𝑋})))
9 funfvex 5431 . . . 4 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝐺𝑋) ∈ V)
109funfni 5218 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) ∈ V)
11 dffv3g 5410 . . 3 ((𝐺𝑋) ∈ V → (𝐹‘(𝐺𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
1210, 11syl 14 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹‘(𝐺𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺𝑋)})))
136, 8, 123eqtr4d 2180 1 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  Vcvv 2681  {csn 3522  cima 4537  ccom 4538  cio 5081   Fn wfn 5113  cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126
This theorem is referenced by:  fvco  5484  fvco3  5485  ofco  5993  updjudhcoinlf  6958  updjudhcoinrg  6959  updjud  6960  caseinl  6969  caseinr  6970  ctm  6987  enomnilem  7003
  Copyright terms: Public domain W3C validator