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Theorem mpt2xopxnop0 7301
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
mpt2xopn0yelv.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
Assertion
Ref Expression
mpt2xopxnop0 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 mpt2xopxnop0
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 neq0 3912 . . 3 (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾))
2 mpt2xopn0yelv.f . . . . . . 7 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
32dmmpt2ssx 7195 . . . . . 6 dom 𝐹 𝑥 ∈ V ({𝑥} × (1st𝑥))
4 elfvdm 6187 . . . . . . 7 (𝑥 ∈ (𝐹‘⟨𝑉, 𝐾⟩) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
5 df-ov 6618 . . . . . . 7 (𝑉𝐹𝐾) = (𝐹‘⟨𝑉, 𝐾⟩)
64, 5eleq2s 2716 . . . . . 6 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
73, 6sseldi 3586 . . . . 5 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)))
8 fveq2 6158 . . . . . . 7 (𝑥 = 𝑉 → (1st𝑥) = (1st𝑉))
98opeliunxp2 5230 . . . . . 6 (⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st𝑉)))
10 eluni 4412 . . . . . . . . 9 (𝐾 dom {𝑉} ↔ ∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}))
11 ne0i 3903 . . . . . . . . . . . . 13 (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅)
1211ad2antlr 762 . . . . . . . . . . . 12 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅)
13 dmsnn0 5569 . . . . . . . . . . . 12 (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅)
1412, 13sylibr 224 . . . . . . . . . . 11 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V))
1514ex 450 . . . . . . . . . 10 ((𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1615exlimiv 1855 . . . . . . . . 9 (∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1710, 16sylbi 207 . . . . . . . 8 (𝐾 dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
18 1stval 7130 . . . . . . . 8 (1st𝑉) = dom {𝑉}
1917, 18eleq2s 2716 . . . . . . 7 (𝐾 ∈ (1st𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
2019impcom 446 . . . . . 6 ((𝑉 ∈ V ∧ 𝐾 ∈ (1st𝑉)) → 𝑉 ∈ (V × V))
219, 20sylbi 207 . . . . 5 (⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) → 𝑉 ∈ (V × V))
227, 21syl 17 . . . 4 (𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
2322exlimiv 1855 . . 3 (∃𝑥 𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
241, 23sylbi 207 . 2 (¬ (𝑉𝐹𝐾) = ∅ → 𝑉 ∈ (V × V))
2524con1i 144 1 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  Vcvv 3190  c0 3897  {csn 4155  cop 4161   cuni 4409   ciun 4492   × cxp 5082  dom cdm 5084  cfv 5857  (class class class)co 6615  cmpt2 6617  1st c1st 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129
This theorem is referenced by:  mpt2xopx0ov0  7302  mpt2xopxprcov0  7303
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