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Theorem mpt2xopn0yelv 5776
 Description: If there is an element of the value 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, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
Hypothesis
Ref Expression
mpt2xopn0yelv.f 𝐹 = (x V, y (1stx) ↦ 𝐶)
Assertion
Ref Expression
mpt2xopn0yelv ((𝑉 𝑋 𝑊 𝑌) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) → 𝐾 𝑉))
Distinct variable groups:   x,y   x,𝐾   x,𝑉   x,𝑊
Allowed substitution hints:   𝐶(x,y)   𝐹(x,y)   𝐾(y)   𝑁(x,y)   𝑉(y)   𝑊(y)   𝑋(x,y)   𝑌(x,y)

Proof of Theorem mpt2xopn0yelv
StepHypRef Expression
1 mpt2xopn0yelv.f . . . . 5 𝐹 = (x V, y (1stx) ↦ 𝐶)
21dmmpt2ssx 5748 . . . 4 dom 𝐹 x V ({x} × (1stx))
31mpt2fun 5526 . . . . . . 7 Fun 𝐹
4 funrel 4845 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 7 . . . . . 6 Rel 𝐹
6 relelfvdm 5130 . . . . . 6 ((Rel 𝐹 𝑁 (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩)) → ⟨⟨𝑉, 𝑊⟩, 𝐾 dom 𝐹)
75, 6mpan 402 . . . . 5 (𝑁 (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩) → ⟨⟨𝑉, 𝑊⟩, 𝐾 dom 𝐹)
8 df-ov 5439 . . . . 5 (⟨𝑉, 𝑊𝐹𝐾) = (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩)
97, 8eleq2s 2114 . . . 4 (𝑁 (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾 dom 𝐹)
102, 9sseldi 2920 . . 3 (𝑁 (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾 x V ({x} × (1stx)))
11 fveq2 5103 . . . . 5 (x = ⟨𝑉, 𝑊⟩ → (1stx) = (1st ‘⟨𝑉, 𝑊⟩))
1211opeliunxp2 4403 . . . 4 (⟨⟨𝑉, 𝑊⟩, 𝐾 x V ({x} × (1stx)) ↔ (⟨𝑉, 𝑊 V 𝐾 (1st ‘⟨𝑉, 𝑊⟩)))
1312simprbi 260 . . 3 (⟨⟨𝑉, 𝑊⟩, 𝐾 x V ({x} × (1stx)) → 𝐾 (1st ‘⟨𝑉, 𝑊⟩))
1410, 13syl 14 . 2 (𝑁 (⟨𝑉, 𝑊𝐹𝐾) → 𝐾 (1st ‘⟨𝑉, 𝑊⟩))
15 op1stg 5700 . . 3 ((𝑉 𝑋 𝑊 𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
1615eleq2d 2089 . 2 ((𝑉 𝑋 𝑊 𝑌) → (𝐾 (1st ‘⟨𝑉, 𝑊⟩) ↔ 𝐾 𝑉))
1714, 16syl5ib 143 1 ((𝑉 𝑋 𝑊 𝑌) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) → 𝐾 𝑉))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  Vcvv 2535  {csn 3350  ⟨cop 3353  ∪ ciun 3631   × cxp 4270  dom cdm 4272  Rel wrel 4277  Fun wfun 4823  ‘cfv 4829  (class class class)co 5436   ↦ cmpt2 5438  1st c1st 5688 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691 This theorem is referenced by:  mpt2xopovel  5778
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