Theorem mpoxneldm 8214

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 3067	. . . 4 ⊢ (𝑋 ∉ 𝐶 ↔ ¬ 𝑋 ∈ 𝐶)
2		df-nel 3067	. . . 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 4306	. . . 4 ⊢ (¬ (𝑋𝐹𝑌) = ∅ ↔ ∃𝑛 𝑛 ∈ (𝑋𝐹𝑌))
7		mpoxeldm.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8	7	mpoxeldm 8213	. . . . 5 ⊢ (𝑛 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
9	8	exlimiv 1963	. . . 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 1570  ∃wex 1812   ∈ wcel 2146   ∉ wnel 3066  ⦋csb 3854  ∅c0 4286  (class class class)co 7419   ∈ cmpo 7421

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993

This theorem is used by:  nbgrnvtx0  29749  clnbgrnvtx0  48652