Theorem mpoxopxnop0 8031

Description: If the first 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 an ordered pair, 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
mpoxopxnop0	⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)

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

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

Proof of Theorem mpoxopxnop0

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 4279	. . 3 ⊢ (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾))
2		mpoxopn0yelv.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)
3	2	dmmpossx 7906	. . . . . 6 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))
4		elfvdm 6806	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹‘⟨𝑉, 𝐾⟩) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
5		df-ov 7278	. . . . . . 7 ⊢ (𝑉𝐹𝐾) = (𝐹‘⟨𝑉, 𝐾⟩)
6	4, 5	eleq2s 2857	. . . . . 6 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
7	3, 6	sselid 3919	. . . . 5 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)))
8		fveq2 6774	. . . . . . 7 ⊢ (𝑥 = 𝑉 → (1st ‘𝑥) = (1st ‘𝑉))
9	8	opeliunxp2 5747	. . . . . 6 ⊢ (⟨𝑉, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉)))
10		eluni 4842	. . . . . . . . 9 ⊢ (𝐾 ∈ ∪ dom {𝑉} ↔ ∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}))
11		ne0i 4268	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅)
12	11	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅)
13		dmsnn0 6110	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅)
14	12, 13	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V))
15	14	ex 413	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
16	15	exlimiv 1933	. . . . . . . . 9 ⊢ (∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
17	10, 16	sylbi 216	. . . . . . . 8 ⊢ (𝐾 ∈ ∪ dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
18		1stval 7833	. . . . . . . 8 ⊢ (1st ‘𝑉) = ∪ dom {𝑉}
19	17, 18	eleq2s 2857	. . . . . . 7 ⊢ (𝐾 ∈ (1st ‘𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
20	19	impcom 408	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉)) → 𝑉 ∈ (V × V))
21	9, 20	sylbi 216	. . . . 5 ⊢ (⟨𝑉, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) → 𝑉 ∈ (V × V))
22	7, 21	syl 17	. . . 4 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
23	22	exlimiv 1933	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
24	1, 23	sylbi 216	. 2 ⊢ (¬ (𝑉𝐹𝐾) = ∅ → 𝑉 ∈ (V × V))
25	24	con1i 147	1 ⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432  ∅c0 4256  {csn 4561  ⟨cop 4567  ∪ cuni 4839  ∪ ciun 4924   × cxp 5587  dom cdm 5589  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  1^st c1st 7829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832

This theorem is referenced by:  mpoxopx0ov0  8032  mpoxopxprcov0  8033