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Theorem fvconst0ci 47713
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 6160 . . . . . 6 dom (𝐴 × {𝐵}) ⊆ 𝐴
32sseli 3970 . . . . 5 (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋𝐴)
4 fvconst0ci.1 . . . . . 6 𝐵 ∈ V
54fvconst2 7197 . . . . 5 (𝑋𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
63, 5syl 17 . . . 4 (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
71, 6eqtrid 2776 . . 3 (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = 𝐵)
87olcd 871 . 2 (𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
9 ndmfv 6916 . . . 4 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅)
101, 9eqtrid 2776 . . 3 𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = ∅)
1110orcd 870 . 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 844   = wceq 1533  wcel 2098  Vcvv 3466  c0 4314  {csn 4620   × cxp 5664  dom cdm 5666  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541
This theorem is referenced by:  f1omo  47715
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