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Theorem mpt2xopovel 5797
Description: 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. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
Hypothesis
Ref Expression
mpt2xopoveq.f 𝐹 = (x V, y (1stx) ↦ {𝑛 (1stx) ∣ φ})
Assertion
Ref Expression
mpt2xopovel ((𝑉 𝑋 𝑊 𝑌) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾 𝑉 𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)))
Distinct variable groups:   𝑛,𝐾,x,y   𝑛,𝑉,x,y   𝑛,𝑊,x,y   𝑛,𝑋,x,y   𝑛,𝑌,x,y   x,𝑁,y
Allowed substitution hints:   φ(x,y,𝑛)   𝐹(x,y,𝑛)   𝑁(𝑛)

Proof of Theorem mpt2xopovel
StepHypRef Expression
1 mpt2xopoveq.f . . . 4 𝐹 = (x V, y (1stx) ↦ {𝑛 (1stx) ∣ φ})
21mpt2xopn0yelv 5795 . . 3 ((𝑉 𝑋 𝑊 𝑌) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) → 𝐾 𝑉))
32pm4.71rd 374 . 2 ((𝑉 𝑋 𝑊 𝑌) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾 𝑉 𝑁 (⟨𝑉, 𝑊𝐹𝐾))))
41mpt2xopoveq 5796 . . . . . 6 (((𝑉 𝑋 𝑊 𝑌) 𝐾 𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛 𝑉[𝑉, 𝑊⟩ / x][𝐾 / y]φ})
54eleq2d 2104 . . . . 5 (((𝑉 𝑋 𝑊 𝑌) 𝐾 𝑉) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) ↔ 𝑁 {𝑛 𝑉[𝑉, 𝑊⟩ / x][𝐾 / y]φ}))
6 nfcv 2175 . . . . . . 7 𝑛𝑉
76elrabsf 2795 . . . . . 6 (𝑁 {𝑛 𝑉[𝑉, 𝑊⟩ / x][𝐾 / y]φ} ↔ (𝑁 𝑉 [𝑁 / 𝑛][𝑉, 𝑊⟩ / x][𝐾 / y]φ))
8 sbccom 2827 . . . . . . . 8 ([𝑁 / 𝑛][𝑉, 𝑊⟩ / x][𝐾 / y]φ[𝑉, 𝑊⟩ / x][𝑁 / 𝑛][𝐾 / y]φ)
9 sbccom 2827 . . . . . . . . 9 ([𝑁 / 𝑛][𝐾 / y]φ[𝐾 / y][𝑁 / 𝑛]φ)
109sbcbii 2812 . . . . . . . 8 ([𝑉, 𝑊⟩ / x][𝑁 / 𝑛][𝐾 / y]φ[𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)
118, 10bitri 173 . . . . . . 7 ([𝑁 / 𝑛][𝑉, 𝑊⟩ / x][𝐾 / y]φ[𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)
1211anbi2i 430 . . . . . 6 ((𝑁 𝑉 [𝑁 / 𝑛][𝑉, 𝑊⟩ / x][𝐾 / y]φ) ↔ (𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ))
137, 12bitri 173 . . . . 5 (𝑁 {𝑛 𝑉[𝑉, 𝑊⟩ / x][𝐾 / y]φ} ↔ (𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ))
145, 13syl6bb 185 . . . 4 (((𝑉 𝑋 𝑊 𝑌) 𝐾 𝑉) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)))
1514pm5.32da 425 . . 3 ((𝑉 𝑋 𝑊 𝑌) → ((𝐾 𝑉 𝑁 (⟨𝑉, 𝑊𝐹𝐾)) ↔ (𝐾 𝑉 (𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ))))
16 3anass 888 . . 3 ((𝐾 𝑉 𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ) ↔ (𝐾 𝑉 (𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)))
1715, 16syl6bbr 187 . 2 ((𝑉 𝑋 𝑊 𝑌) → ((𝐾 𝑉 𝑁 (⟨𝑉, 𝑊𝐹𝐾)) ↔ (𝐾 𝑉 𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)))
183, 17bitrd 177 1 ((𝑉 𝑋 𝑊 𝑌) → (𝑁 (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾 𝑉 𝑁 𝑉 [𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  {crab 2304  Vcvv 2551  [wsbc 2758  cop 3370  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-in1 544  ax-in2 545  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-bndl 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  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  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-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  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: (None)
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