Theorem mpoxneldm 8154

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 3037	. . . 4 ⊢ (𝑋 ∉ 𝐶 ↔ ¬ 𝑋 ∈ 𝐶)
2		df-nel 3037	. . . 4 ⊢ (𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷 ↔ ¬ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷)
3	1, 2	orbi12i 914	. . 3 ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) ↔ (¬ 𝑋 ∈ 𝐶 ∨ ¬ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
4		ianor 983	. . 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 4304	. . . 4 ⊢ (¬ (𝑋𝐹𝑌) = ∅ ↔ ∃𝑛 𝑛 ∈ (𝑋𝐹𝑌))
7		mpoxeldm.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8	7	mpoxeldm 8153	. . . . 5 ⊢ (𝑛 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
9	8	exlimiv 1931	. . . 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 847   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ∉ wnel 3036  ⦋csb 3849  ∅c0 4285  (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 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934

This theorem is referenced by:  nbgrnvtx0  29412  clnbgrnvtx0  48073