Theorem mpoxneldm 8237

Description: If the first argument of an operation given by a maps-to rule is not an element of the first component of the domain or the second argument is not an element of the second component of the domain depending on the first argument, then the value of the operation is the empty set. (Contributed by AV, 25-Oct-2020.)

Hypothesis

Ref	Expression
mpoxeldm.f	⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)

Assertion

Ref	Expression
mpoxneldm	⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) → (𝑋𝐹𝑌) = ∅)

Distinct variable groups:   𝑥,𝐶,𝑦   𝑦,𝐷   𝑥,𝑋   𝑥,𝑌

Allowed substitution hints:   𝐷(𝑥)   𝑅(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝑋(𝑦)   𝑌(𝑦)

Proof of Theorem mpoxneldm

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nel 3062	. . . 4 ⊢ (𝑋 ∉ 𝐶 ↔ ¬ 𝑋 ∈ 𝐶)
2		df-nel 3062	. . . 4 ⊢ (𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷 ↔ ¬ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷)
3	1, 2	orbi12i 928	. . 3 ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) ↔ (¬ 𝑋 ∈ 𝐶 ∨ ¬ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
4		ianor 997	. . 3 ⊢ (¬ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷) ↔ (¬ 𝑋 ∈ 𝐶 ∨ ¬ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
5	3, 4	bitr4i 281	. 2 ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) ↔ ¬ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
6		neq0 4298	. . . 4 ⊢ (¬ (𝑋𝐹𝑌) = ∅ ↔ ∃𝑛 𝑛 ∈ (𝑋𝐹𝑌))
7		mpoxeldm.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8	7	mpoxeldm 8236	. . . . 5 ⊢ (𝑛 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
9	8	exlimiv 1962	. . . 4 ⊢ (∃𝑛 𝑛 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
10	6, 9	sylbi 220	. . 3 ⊢ (¬ (𝑋𝐹𝑌) = ∅ → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
11	10	con1i 148	. 2 ⊢ (¬ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷) → (𝑋𝐹𝑌) = ∅)
12	5, 11	sylbi 220	1 ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) → (𝑋𝐹𝑌) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ∉ wnel 3061  ⦋csb 3847  ∅c0 4277  (class class class)co 7424   ∈ cmpo 7426

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 5391  ax-un 7755

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-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006

This theorem is used by:  nbgrnvtx0  29929  clnbgrnvtx0  49061