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Theorem fcoconst 5545
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 502 . . 3 (((𝐹 Fn 𝑋𝑌𝑋) ∧ 𝑥𝐼) → 𝑌𝑋)
2 fconstmpt 4546 . . . 4 (𝐼 × {𝑌}) = (𝑥𝐼𝑌)
32a1i 9 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐼 × {𝑌}) = (𝑥𝐼𝑌))
4 simpl 108 . . . . 5 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 Fn 𝑋)
5 dffn2 5232 . . . . 5 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
64, 5sylib 121 . . . 4 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹:𝑋⟶V)
76feqmptd 5428 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 = (𝑦𝑋 ↦ (𝐹𝑦)))
8 fveq2 5375 . . 3 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
91, 3, 7, 8fmptco 5540 . 2 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥𝐼 ↦ (𝐹𝑌)))
10 fconstmpt 4546 . 2 (𝐼 × {(𝐹𝑌)}) = (𝑥𝐼 ↦ (𝐹𝑌))
119, 10syl6eqr 2165 1 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  Vcvv 2657  {csn 3493  cmpt 3949   × cxp 4497  ccom 4503   Fn wfn 5076  wf 5077  cfv 5081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089
This theorem is referenced by: (None)
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