Theorem mpoxopynvov0 8202

Description: If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Hypothesis

Ref	Expression
mpoxopn0yelv.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)

Assertion

Ref	Expression
mpoxopynvov0	⊢ (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐾   𝑥,𝑉   𝑥,𝑊   𝑥,𝐹

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑦)   𝐾(𝑦)   𝑉(𝑦)   𝑊(𝑦)

Proof of Theorem mpoxopynvov0

Step	Hyp	Ref	Expression
1		mpoxopn0yelv.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)
2	1	mpoxopynvov0g 8198	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝑊 ∈ V) ∧ 𝐾 ∉ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
3	2	ex 417	. 2 ⊢ ((𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅))
4	1	mpoxopxprcov0 8201	. . 3 ⊢ (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
5	4	a1d 26	. 2 ⊢ (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅))
6	3, 5	pm2.61i 184	1 ⊢ (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ∉ wnel 3064  Vcvv 3457  ∅c0 4288  ⟨cop 4591  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  1^st c1st 7972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975

This theorem is referenced by:  mpoxopoveqd  8205