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Theorem fvconst0ci 48689
Description: A constant function's value is either the constant or the empty set. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
Hypotheses
Ref Expression
fvconst0ci.1 𝐵 ∈ V
fvconst0ci.2 𝑌 = ((𝐴 × {𝐵})‘𝑋)
Assertion
Ref Expression
fvconst0ci (𝑌 = ∅ ∨ 𝑌 = 𝐵)

Proof of Theorem fvconst0ci
StepHypRef Expression
1 fvconst0ci.2 . . . 4 𝑌 = ((𝐴 × {𝐵})‘𝑋)
2 dmxpss 6193 . . . . . 6 dom (𝐴 × {𝐵}) ⊆ 𝐴
32sseli 3991 . . . . 5 (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋𝐴)
4 fvconst0ci.1 . . . . . 6 𝐵 ∈ V
54fvconst2 7224 . . . . 5 (𝑋𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
63, 5syl 17 . . . 4 (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
71, 6eqtrid 2787 . . 3 (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = 𝐵)
87olcd 874 . 2 (𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
9 ndmfv 6942 . . . 4 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅)
101, 9eqtrid 2787 . . 3 𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = ∅)
1110orcd 873 . 2 𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
128, 11pm2.61i 182 1 (𝑌 = ∅ ∨ 𝑌 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 847   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {csn 4631   × cxp 5687  dom cdm 5689  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  f1omo  48691
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