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Theorem fvco4i 6985
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 6571 . . . 4 (Fun 𝐺𝐺 Fn dom 𝐺)
31, 2mpbi 229 . . 3 𝐺 Fn dom 𝐺
4 fvco2 6981 . . 3 ((𝐺 Fn dom 𝐺𝑋 ∈ dom 𝐺) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
53, 4mpan 687 . 2 (𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
6 fvco4i.a . . 3 ∅ = (𝐹‘∅)
7 dmcoss 5963 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐺
87sseli 3973 . . . 4 (𝑋 ∈ dom (𝐹𝐺) → 𝑋 ∈ dom 𝐺)
9 ndmfv 6919 . . . 4 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)‘𝑋) = ∅)
108, 9nsyl5 159 . . 3 𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = ∅)
11 ndmfv 6919 . . . 4 𝑋 ∈ dom 𝐺 → (𝐺𝑋) = ∅)
1211fveq2d 6888 . . 3 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺𝑋)) = (𝐹‘∅))
136, 10, 123eqtr4a 2792 . 2 𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
145, 13pm2.61i 182 1 ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  c0 4317  dom cdm 5669  ccom 5673  Fun wfun 6530   Fn wfn 6531  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544
This theorem is referenced by:  lidlval  21067  rspval  21068
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