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Theorem fvconst0ci 47365
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 6162 . . . . . 6 dom (𝐴 × {𝐵}) ⊆ 𝐴
32sseli 3976 . . . . 5 (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋𝐴)
4 fvconst0ci.1 . . . . . 6 𝐵 ∈ V
54fvconst2 7192 . . . . 5 (𝑋𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
63, 5syl 17 . . . 4 (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
71, 6eqtrid 2785 . . 3 (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = 𝐵)
87olcd 873 . 2 (𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
9 ndmfv 6916 . . . 4 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅)
101, 9eqtrid 2785 . . 3 𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = ∅)
1110orcd 872 . 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 846   = wceq 1542  wcel 2107  Vcvv 3475  c0 4320  {csn 4624   × cxp 5670  dom cdm 5672  cfv 6535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543
This theorem is referenced by:  f1omo  47367
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