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Theorem fvconstdomi 45757
Description: A constant function's value is dominated by the constant. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
Hypothesis
Ref Expression
fvconstdomi.1 𝐵 ∈ V
Assertion
Ref Expression
fvconstdomi ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵

Proof of Theorem fvconstdomi
StepHypRef Expression
1 dmxpss 6013 . . . . 5 dom (𝐴 × {𝐵}) ⊆ 𝐴
21sseli 3883 . . . 4 (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋𝐴)
3 fvconstdomi.1 . . . . 5 𝐵 ∈ V
43fvconst2 6988 . . . 4 (𝑋𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
52, 4syl 17 . . 3 (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
6 domrefg 8602 . . . 4 (𝐵 ∈ V → 𝐵𝐵)
73, 6ax-mp 5 . . 3 𝐵𝐵
85, 7eqbrtrdi 5079 . 2 (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵)
9 ndmfv 6716 . . 3 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅)
1030dom 8709 . . 3 ∅ ≼ 𝐵
119, 10eqbrtrdi 5079 . 2 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵)
128, 11pm2.61i 185 1 ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2114  Vcvv 3400  c0 4221  {csn 4526   class class class wbr 5040   × cxp 5533  dom cdm 5535  cfv 6349  cdom 8565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-en 8568  df-dom 8569
This theorem is referenced by:  f1omoALT  45759
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