Theorem fvconst0ci 50099

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 6162	. . . . . 6 ⊢ dom (𝐴 × {𝐵}) ⊆ 𝐴
3	2	sseli 3904	. . . . 5 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋 ∈ 𝐴)
4		fvconst0ci.1	. . . . . 6 ⊢ 𝐵 ∈ V
5	4	fvconst2 7211	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
6	3, 5	syl 18	. . . 4 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
7	1, 6	eqtrid 2807	. . 3 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = 𝐵)
8	7	olcd 888	. 2 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
9		ndmfv 6918	. . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅)
10	1, 9	eqtrid 2807	. . 3 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = ∅)
11	10	orcd 887	. 2 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵))
12	8, 11	pm2.61i 184	1 ⊢ (𝑌 = ∅ ∨ 𝑌 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ∅c0 4278  {csn 4584   × cxp 5648  dom cdm 5650  ‘cfv 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547

This theorem is used by:  f1omo  50101  f1omoOLD  50102