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Theorem fcoconst 6356
 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 791 . . 3 (((𝐹 Fn 𝑋𝑌𝑋) ∧ 𝑥𝐼) → 𝑌𝑋)
2 fconstmpt 5128 . . . 4 (𝐼 × {𝑌}) = (𝑥𝐼𝑌)
32a1i 11 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐼 × {𝑌}) = (𝑥𝐼𝑌))
4 simpl 473 . . . . 5 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 Fn 𝑋)
5 dffn2 6006 . . . . 5 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
64, 5sylib 208 . . . 4 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹:𝑋⟶V)
76feqmptd 6207 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 = (𝑦𝑋 ↦ (𝐹𝑦)))
8 fveq2 6150 . . 3 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
91, 3, 7, 8fmptco 6352 . 2 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥𝐼 ↦ (𝐹𝑌)))
10 fconstmpt 5128 . 2 (𝐼 × {(𝐹𝑌)}) = (𝑥𝐼 ↦ (𝐹𝑌))
119, 10syl6eqr 2678 1 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1992  Vcvv 3191  {csn 4153   ↦ cmpt 4678   × cxp 5077   ∘ ccom 5083   Fn wfn 5845  ⟶wf 5846  ‘cfv 5850 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858 This theorem is referenced by:  s1co  13511  setcmon  16653  pwsco2mhm  17287  pws1  18532  pwsmgp  18534  imasdsf1olem  22083  cvmliftphtlem  30999  cvmlift3lem9  31009
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