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Theorem fvco4i 6315
Description: Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Hypotheses
Ref Expression
fvco4i.a ∅ = (𝐹‘∅)
fvco4i.b Fun 𝐺
Assertion
Ref Expression
fvco4i ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋))

Proof of Theorem fvco4i
StepHypRef Expression
1 fvco4i.b . . . 4 Fun 𝐺
2 funfn 5956 . . . 4 (Fun 𝐺𝐺 Fn dom 𝐺)
31, 2mpbi 220 . . 3 𝐺 Fn dom 𝐺
4 fvco2 6312 . . 3 ((𝐺 Fn dom 𝐺𝑋 ∈ dom 𝐺) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
53, 4mpan 706 . 2 (𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
6 fvco4i.a . . 3 ∅ = (𝐹‘∅)
7 dmcoss 5417 . . . . . 6 dom (𝐹𝐺) ⊆ dom 𝐺
87sseli 3632 . . . . 5 (𝑋 ∈ dom (𝐹𝐺) → 𝑋 ∈ dom 𝐺)
98con3i 150 . . . 4 𝑋 ∈ dom 𝐺 → ¬ 𝑋 ∈ dom (𝐹𝐺))
10 ndmfv 6256 . . . 4 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)‘𝑋) = ∅)
119, 10syl 17 . . 3 𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = ∅)
12 ndmfv 6256 . . . 4 𝑋 ∈ dom 𝐺 → (𝐺𝑋) = ∅)
1312fveq2d 6233 . . 3 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺𝑋)) = (𝐹‘∅))
146, 11, 133eqtr4a 2711 . 2 𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
155, 14pm2.61i 176 1 ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wcel 2030  c0 3948  dom cdm 5143  ccom 5147  Fun wfun 5920   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  lidlval  19240  rspval  19241
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