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Theorem mpt2xopxnop0 7511
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 4073 . . 3 (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾))
2 mpt2xopn0yelv.f . . . . . . 7 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
32dmmpt2ssx 7404 . . . . . 6 dom 𝐹 𝑥 ∈ V ({𝑥} × (1st𝑥))
4 elfvdm 6382 . . . . . . 7 (𝑥 ∈ (𝐹‘⟨𝑉, 𝐾⟩) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
5 df-ov 6817 . . . . . . 7 (𝑉𝐹𝐾) = (𝐹‘⟨𝑉, 𝐾⟩)
64, 5eleq2s 2857 . . . . . 6 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
73, 6sseldi 3742 . . . . 5 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)))
8 fveq2 6353 . . . . . . 7 (𝑥 = 𝑉 → (1st𝑥) = (1st𝑉))
98opeliunxp2 5416 . . . . . 6 (⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st𝑉)))
10 eluni 4591 . . . . . . . . 9 (𝐾 dom {𝑉} ↔ ∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}))
11 ne0i 4064 . . . . . . . . . . . . 13 (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅)
1211ad2antlr 765 . . . . . . . . . . . 12 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅)
13 dmsnn0 5758 . . . . . . . . . . . 12 (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅)
1412, 13sylibr 224 . . . . . . . . . . 11 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V))
1514ex 449 . . . . . . . . . 10 ((𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1615exlimiv 2007 . . . . . . . . 9 (∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1710, 16sylbi 207 . . . . . . . 8 (𝐾 dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
18 1stval 7336 . . . . . . . 8 (1st𝑉) = dom {𝑉}
1917, 18eleq2s 2857 . . . . . . 7 (𝐾 ∈ (1st𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
2019impcom 445 . . . . . 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 2007 . . 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 383   = wceq 1632  wex 1853  wcel 2139  wne 2932  Vcvv 3340  c0 4058  {csn 4321  cop 4327   cuni 4588   ciun 4672   × cxp 5264  dom cdm 5266  cfv 6049  (class class class)co 6814  cmpt2 6816  1st c1st 7332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335
This theorem is referenced by:  mpt2xopx0ov0  7512  mpt2xopxprcov0  7513
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