Theorem fvconst0ci 49817

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 6164	. . . . . 6 ⊢ dom (𝐴 × {𝐵}) ⊆ 𝐴
3	2	sseli 3927	. . . . 5 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋 ∈ 𝐴)
4		fvconst0ci.1	. . . . . 6 ⊢ 𝐵 ∈ V
5	4	fvconst2 7203	. . . . 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 6910	. . . 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 1570   ∈ wcel 2145  Vcvv 3450  ∅c0 4279  {csn 4584   × cxp 5653  dom cdm 5655  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541

This theorem is used by:  f1omo  49819  f1omoOLD  49820