Theorem fvconst0ci 46186

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

Step	Hyp	Ref	Expression
1		fvconst0ci.2	. . . 4 ⊢ 𝑌 = ((𝐴 × {𝐵})‘𝑋)
2		dmxpss 6074	. . . . . 6 ⊢ dom (𝐴 × {𝐵}) ⊆ 𝐴
3	2	sseli 3917	. . . . 5 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋 ∈ 𝐴)
4		fvconst0ci.1	. . . . . 6 ⊢ 𝐵 ∈ V
5	4	fvconst2 7079	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
6	3, 5	syl 17	. . . 4 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
7	1, 6	eqtrid 2790	. . 3 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = 𝐵)
8	7	olcd 871	. 2 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
9		ndmfv 6804	. . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅)
10	1, 9	eqtrid 2790	. . 3 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = ∅)
11	10	orcd 870	. 2 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
12	8, 11	pm2.61i 182	1 ⊢ (𝑌 = ∅ ∨ 𝑌 = 𝐵)

Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  {csn 4561   × cxp 5587  dom cdm 5589  ‘cfv 6433

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  f1omo  46188