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Theorem mpoxopoveq 7872
Description: 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, 11-Oct-2017.)
Hypothesis
Ref Expression
mpoxopoveq.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
Assertion
Ref Expression
mpoxopoveq (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Distinct variable groups:   𝑛,𝐾,𝑥,𝑦   𝑛,𝑉,𝑥,𝑦   𝑛,𝑊,𝑥,𝑦   𝑛,𝑋,𝑥,𝑦   𝑛,𝑌,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑛)   𝐹(𝑥,𝑦,𝑛)

Proof of Theorem mpoxopoveq
StepHypRef Expression
1 mpoxopoveq.f . . 3 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
21a1i 11 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑}))
3 fveq2 6649 . . . . 5 (𝑥 = ⟨𝑉, 𝑊⟩ → (1st𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
4 op1stg 7687 . . . . . 6 ((𝑉𝑋𝑊𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
54adantr 484 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
63, 5sylan9eqr 2858 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ 𝑥 = ⟨𝑉, 𝑊⟩) → (1st𝑥) = 𝑉)
76adantrr 716 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (1st𝑥) = 𝑉)
8 sbceq1a 3734 . . . . . 6 (𝑦 = 𝐾 → (𝜑[𝐾 / 𝑦]𝜑))
98adantl 485 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → (𝜑[𝐾 / 𝑦]𝜑))
109adantl 485 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝐾 / 𝑦]𝜑))
11 sbceq1a 3734 . . . . . 6 (𝑥 = ⟨𝑉, 𝑊⟩ → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1211adantr 484 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1312adantl 485 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1410, 13bitrd 282 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
157, 14rabeqbidv 3436 . 2 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → {𝑛 ∈ (1st𝑥) ∣ 𝜑} = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
16 opex 5324 . . 3 𝑉, 𝑊⟩ ∈ V
1716a1i 11 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → ⟨𝑉, 𝑊⟩ ∈ V)
18 simpr 488 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → 𝐾𝑉)
19 rabexg 5201 . . 3 (𝑉𝑋 → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
2019ad2antrr 725 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
21 equid 2019 . . 3 𝑧 = 𝑧
22 nfvd 1916 . . 3 (𝑧 = 𝑧 → Ⅎ𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2321, 22ax-mp 5 . 2 𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
24 nfvd 1916 . . 3 (𝑧 = 𝑧 → Ⅎ𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2521, 24ax-mp 5 . 2 𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
26 nfcv 2958 . 2 𝑦𝑉, 𝑊
27 nfcv 2958 . 2 𝑥𝐾
28 nfsbc1v 3743 . . 3 𝑥[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
29 nfcv 2958 . . 3 𝑥𝑉
3028, 29nfrabw 3341 . 2 𝑥{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
31 nfsbc1v 3743 . . . 4 𝑦[𝐾 / 𝑦]𝜑
3226, 31nfsbcw 3745 . . 3 𝑦[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
33 nfcv 2958 . . 3 𝑦𝑉
3432, 33nfrabw 3341 . 2 𝑦{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpodxf 7283 1 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2112  {crab 3113  Vcvv 3444  [wsbc 3723  cop 4534  cfv 6328  (class class class)co 7139  cmpo 7141  1st c1st 7673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675
This theorem is referenced by:  mpoxopovel  7873  mpoxopoveqd  7874
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