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Theorem fvco4i 6583
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 6212 . . . 4 (Fun 𝐺𝐺 Fn dom 𝐺)
31, 2mpbi 222 . . 3 𝐺 Fn dom 𝐺
4 fvco2 6580 . . 3 ((𝐺 Fn dom 𝐺𝑋 ∈ dom 𝐺) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
53, 4mpan 677 . 2 (𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
6 fvco4i.a . . 3 ∅ = (𝐹‘∅)
7 dmcoss 5677 . . . . . 6 dom (𝐹𝐺) ⊆ dom 𝐺
87sseli 3850 . . . . 5 (𝑋 ∈ dom (𝐹𝐺) → 𝑋 ∈ dom 𝐺)
98con3i 152 . . . 4 𝑋 ∈ dom 𝐺 → ¬ 𝑋 ∈ dom (𝐹𝐺))
10 ndmfv 6523 . . . 4 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)‘𝑋) = ∅)
119, 10syl 17 . . 3 𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = ∅)
12 ndmfv 6523 . . . 4 𝑋 ∈ dom 𝐺 → (𝐺𝑋) = ∅)
1312fveq2d 6497 . . 3 𝑋 ∈ dom 𝐺 → (𝐹‘(𝐺𝑋)) = (𝐹‘∅))
146, 11, 133eqtr4a 2834 . 2 𝑋 ∈ dom 𝐺 → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
155, 14pm2.61i 177 1 ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1507  wcel 2048  c0 4173  dom cdm 5400  ccom 5404  Fun wfun 6176   Fn wfn 6177  cfv 6182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-fv 6190
This theorem is referenced by:  lidlval  19676  rspval  19677
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