Theorem fvconst0ci 48790

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 6191	. . . . . 6 ⊢ dom (𝐴 × {𝐵}) ⊆ 𝐴
3	2	sseli 3979	. . . . 5 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋 ∈ 𝐴)
4		fvconst0ci.1	. . . . . 6 ⊢ 𝐵 ∈ V
5	4	fvconst2 7224	. . . . 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 2789	. . 3 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = 𝐵)
8	7	olcd 875	. 2 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
9		ndmfv 6941	. . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅)
10	1, 9	eqtrid 2789	. . 3 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = ∅)
11	10	orcd 874	. 2 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
12	8, 11	pm2.61i 182	1 ⊢ (𝑌 = ∅ ∨ 𝑌 = 𝐵)

Syntax hints:  ¬ wn 3   ∨ wo 848   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  {csn 4626   × cxp 5683  dom cdm 5685  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569

This theorem is referenced by:  f1omo  48792