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Theorem fcoconst 7128
Description: Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
Assertion
Ref Expression
fcoconst ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))

Proof of Theorem fcoconst
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 766 . . 3 (((𝐹 Fn 𝑋𝑌𝑋) ∧ 𝑥𝐼) → 𝑌𝑋)
2 fconstmpt 5731 . . . 4 (𝐼 × {𝑌}) = (𝑥𝐼𝑌)
32a1i 11 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐼 × {𝑌}) = (𝑥𝐼𝑌))
4 simpl 482 . . . . 5 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 Fn 𝑋)
5 dffn2 6713 . . . . 5 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
64, 5sylib 217 . . . 4 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹:𝑋⟶V)
76feqmptd 6954 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 = (𝑦𝑋 ↦ (𝐹𝑦)))
8 fveq2 6885 . . 3 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
91, 3, 7, 8fmptco 7123 . 2 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥𝐼 ↦ (𝐹𝑌)))
10 fconstmpt 5731 . 2 (𝐼 × {(𝐹𝑌)}) = (𝑥𝐼 ↦ (𝐹𝑌))
119, 10eqtr4di 2784 1 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  {csn 4623  cmpt 5224   × cxp 5667  ccom 5673   Fn wfn 6532  wf 6533  cfv 6537
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-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  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-mpt 5225  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 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545
This theorem is referenced by:  f1ofvswap  7300  s1co  14790  setcmon  18049  pwsco2mhm  18758  smndex1igid  18829  pws1  20224  pwsmgp  20226  imasdsf1olem  24234  cvmliftphtlem  34836  cvmlift3lem9  34846
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