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Theorem mpt2xopn0yelv 5795
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 5767 . . . 4 dom 𝐹 x V ({x} × (1stx))
31mpt2fun 5545 . . . . . . 7 Fun 𝐹
4 funrel 4862 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 7 . . . . . 6 Rel 𝐹
6 relelfvdm 5148 . . . . . 6 ((Rel 𝐹 𝑁 (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩)) → ⟨⟨𝑉, 𝑊⟩, 𝐾 dom 𝐹)
75, 6mpan 400 . . . . 5 (𝑁 (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩) → ⟨⟨𝑉, 𝑊⟩, 𝐾 dom 𝐹)
8 df-ov 5458 . . . . 5 (⟨𝑉, 𝑊𝐹𝐾) = (𝐹‘⟨⟨𝑉, 𝑊⟩, 𝐾⟩)
97, 8eleq2s 2129 . . . 4 (𝑁 (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾 dom 𝐹)
102, 9sseldi 2937 . . 3 (𝑁 (⟨𝑉, 𝑊𝐹𝐾) → ⟨⟨𝑉, 𝑊⟩, 𝐾 x V ({x} × (1stx)))
11 fveq2 5121 . . . . 5 (x = ⟨𝑉, 𝑊⟩ → (1stx) = (1st ‘⟨𝑉, 𝑊⟩))
1211opeliunxp2 4419 . . . 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 5719 . . 3 ((𝑉 𝑋 𝑊 𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
1615eleq2d 2104 . 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 1242   wcel 1390  Vcvv 2551  {csn 3367  cop 3370   ciun 3648   × cxp 4286  dom cdm 4288  Rel wrel 4293  Fun wfun 4839  cfv 4845  (class class class)co 5455  cmpt2 5457  1st c1st 5707
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  mpt2xopovel  5797
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