Theorem mpoxneldm 8153

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 3038	. . . 4 ⊢ (𝑋 ∉ 𝐶 ↔ ¬ 𝑋 ∈ 𝐶)
2		df-nel 3038	. . . 4 ⊢ (𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷 ↔ ¬ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷)
3	1, 2	orbi12i 915	. . 3 ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) ↔ (¬ 𝑋 ∈ 𝐶 ∨ ¬ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
4		ianor 984	. . 3 ⊢ (¬ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷) ↔ (¬ 𝑋 ∈ 𝐶 ∨ ¬ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
5	3, 4	bitr4i 278	. 2 ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) ↔ ¬ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
6		neq0 4293	. . . 4 ⊢ (¬ (𝑋𝐹𝑌) = ∅ ↔ ∃𝑛 𝑛 ∈ (𝑋𝐹𝑌))
7		mpoxeldm.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8	7	mpoxeldm 8152	. . . . 5 ⊢ (𝑛 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
9	8	exlimiv 1932	. . . 4 ⊢ (∃𝑛 𝑛 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
10	6, 9	sylbi 217	. . 3 ⊢ (¬ (𝑋𝐹𝑌) = ∅ → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
11	10	con1i 147	. 2 ⊢ (¬ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷) → (𝑋𝐹𝑌) = ∅)
12	5, 11	sylbi 217	1 ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) → (𝑋𝐹𝑌) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ∉ wnel 3037  ⦋csb 3838  ∅c0 4274  (class class class)co 7358   ∈ cmpo 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934

This theorem is referenced by:  nbgrnvtx0  29427  clnbgrnvtx0  48300