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Theorem fvco4i 7010
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 6596 . . . 4 (Fun 𝐺𝐺 Fn dom 𝐺)
31, 2mpbi 230 . . 3 𝐺 Fn dom 𝐺
4 fvco2 7006 . . 3 ((𝐺 Fn dom 𝐺𝑋 ∈ dom 𝐺) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
53, 4mpan 690 . 2 (𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
6 fvco4i.a . . 3 ∅ = (𝐹‘∅)
7 dmcoss 5985 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐺
87sseli 3979 . . . 4 (𝑋 ∈ dom (𝐹𝐺) → 𝑋 ∈ dom 𝐺)
9 ndmfv 6941 . . . 4 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)‘𝑋) = ∅)
108, 9nsyl5 159 . . 3 𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = ∅)
11 ndmfv 6941 . . . 4 𝑋 ∈ dom 𝐺 → (𝐺𝑋) = ∅)
1211fveq2d 6910 . . 3 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺𝑋)) = (𝐹‘∅))
136, 10, 123eqtr4a 2803 . 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 1540  wcel 2108  c0 4333  dom cdm 5685  ccom 5689  Fun wfun 6555   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  lidlval  21220  rspval  21221
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