Theorem fvconst0ci 46074

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 6063	. . . . . 6 ⊢ dom (𝐴 × {𝐵}) ⊆ 𝐴
3	2	sseli 3913	. . . . 5 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋 ∈ 𝐴)
4		fvconst0ci.1	. . . . . 6 ⊢ 𝐵 ∈ V
5	4	fvconst2 7061	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
6	3, 5	syl 17	. . . 4 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
7	1, 6	syl5eq 2791	. . 3 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = 𝐵)
8	7	olcd 870	. 2 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
9		ndmfv 6786	. . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅)
10	1, 9	syl5eq 2791	. . 3 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = ∅)
11	10	orcd 869	. 2 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
12	8, 11	pm2.61i 182	1 ⊢ (𝑌 = ∅ ∨ 𝑌 = 𝐵)

Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  {csn 4558   × cxp 5578  dom cdm 5580  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426

This theorem is referenced by:  f1omo  46076