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Theorem fvco4i 6744
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 6364 . . . 4 (Fun 𝐺𝐺 Fn dom 𝐺)
31, 2mpbi 233 . . 3 𝐺 Fn dom 𝐺
4 fvco2 6740 . . 3 ((𝐺 Fn dom 𝐺𝑋 ∈ dom 𝐺) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
53, 4mpan 689 . 2 (𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
6 fvco4i.a . . 3 ∅ = (𝐹‘∅)
7 dmcoss 5820 . . . . 5 dom (𝐹𝐺) ⊆ dom 𝐺
87sseli 3938 . . . 4 (𝑋 ∈ dom (𝐹𝐺) → 𝑋 ∈ dom 𝐺)
9 ndmfv 6682 . . . 4 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)‘𝑋) = ∅)
108, 9nsyl5 162 . . 3 𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = ∅)
11 ndmfv 6682 . . . 4 𝑋 ∈ dom 𝐺 → (𝐺𝑋) = ∅)
1211fveq2d 6656 . . 3 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺𝑋)) = (𝐹‘∅))
136, 10, 123eqtr4a 2883 . 2 𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
145, 13pm2.61i 185 1 ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2114  c0 4265  dom cdm 5532  ccom 5536  Fun wfun 6328   Fn wfn 6329  cfv 6334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342
This theorem is referenced by:  lidlval  19955  rspval  19956
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