Theorem mpoxopxnop0 8213

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 4299	. . 3 ⊢ (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾))
2		mpoxopn0yelv.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)
3	2	dmmpossx 8063	. . . . . 6 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))
4		elfvdm 6912	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹‘⟨𝑉, 𝐾⟩) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
5		df-ov 7416	. . . . . . 7 ⊢ (𝑉𝐹𝐾) = (𝐹‘⟨𝑉, 𝐾⟩)
6	4, 5	eleq2s 2878	. . . . . 6 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
7	3, 6	sselid 3929	. . . . 5 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)))
8		fveq2 6878	. . . . . . 7 ⊢ (𝑥 = 𝑉 → (1st ‘𝑥) = (1st ‘𝑉))
9	8	opeliunxp2 5818	. . . . . 6 ⊢ (⟨𝑉, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉)))
10		eluni 4870	. . . . . . . . 9 ⊢ (𝐾 ∈ ∪ dom {𝑉} ↔ ∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}))
11		ne0i 4287	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅)
12	11	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅)
13		dmsnn0 6203	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅)
14	12, 13	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V))
15	14	ex 418	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
16	15	exlimiv 1963	. . . . . . . . 9 ⊢ (∃𝑛(𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
17	10, 16	sylbi 220	. . . . . . . 8 ⊢ (𝐾 ∈ ∪ dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
18		1stval 7988	. . . . . . . 8 ⊢ (1st ‘𝑉) = ∪ dom {𝑉}
19	17, 18	eleq2s 2878	. . . . . . 7 ⊢ (𝐾 ∈ (1st ‘𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
20	19	impcom 413	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐾 ∈ (1st ‘𝑉)) → 𝑉 ∈ (V × V))
21	9, 20	sylbi 220	. . . . 5 ⊢ (⟨𝑉, 𝐾⟩ ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) → 𝑉 ∈ (V × V))
22	7, 21	syl 18	. . . 4 ⊢ (𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
23	22	exlimiv 1963	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
24	1, 23	sylbi 220	. 2 ⊢ (¬ (𝑉𝐹𝐾) = ∅ → 𝑉 ∈ (V × V))
25	24	con1i 148	1 ⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  Vcvv 3450  ∅c0 4279  {csn 4584  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951   × cxp 5653  dom cdm 5655  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7984

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987

This theorem is used by:  mpoxopx0ov0  8214  mpoxopxprcov0  8215