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Theorem mpt2xopoveq 7330
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
mpt2xopoveq.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
Assertion
Ref Expression
mpt2xopoveq (((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 mpt2xopoveq
StepHypRef Expression
1 mpt2xopoveq.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 6178 . . . . 5 (𝑥 = ⟨𝑉, 𝑊⟩ → (1st𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
4 op1stg 7165 . . . . . 6 ((𝑉𝑋𝑊𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
54adantr 481 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
63, 5sylan9eqr 2676 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ 𝑥 = ⟨𝑉, 𝑊⟩) → (1st𝑥) = 𝑉)
76adantrr 752 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (1st𝑥) = 𝑉)
8 sbceq1a 3440 . . . . . 6 (𝑦 = 𝐾 → (𝜑[𝐾 / 𝑦]𝜑))
98adantl 482 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → (𝜑[𝐾 / 𝑦]𝜑))
109adantl 482 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝐾 / 𝑦]𝜑))
11 sbceq1a 3440 . . . . . 6 (𝑥 = ⟨𝑉, 𝑊⟩ → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1211adantr 481 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1312adantl 482 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1410, 13bitrd 268 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
157, 14rabeqbidv 3190 . 2 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → {𝑛 ∈ (1st𝑥) ∣ 𝜑} = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
16 opex 4923 . . 3 𝑉, 𝑊⟩ ∈ V
1716a1i 11 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → ⟨𝑉, 𝑊⟩ ∈ V)
18 simpr 477 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → 𝐾𝑉)
19 rabexg 4803 . . 3 (𝑉𝑋 → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
2019ad2antrr 761 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
21 equid 1937 . . 3 𝑧 = 𝑧
22 nfvd 1842 . . 3 (𝑧 = 𝑧 → Ⅎ𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2321, 22ax-mp 5 . 2 𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
24 nfvd 1842 . . 3 (𝑧 = 𝑧 → Ⅎ𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2521, 24ax-mp 5 . 2 𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
26 nfcv 2762 . 2 𝑦𝑉, 𝑊
27 nfcv 2762 . 2 𝑥𝐾
28 nfsbc1v 3449 . . 3 𝑥[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
29 nfcv 2762 . . 3 𝑥𝑉
3028, 29nfrab 3118 . 2 𝑥{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
31 nfsbc1v 3449 . . . 4 𝑦[𝐾 / 𝑦]𝜑
3226, 31nfsbc 3451 . . 3 𝑦[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
33 nfcv 2762 . . 3 𝑦𝑉
3432, 33nfrab 3118 . 2 𝑦{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 6771 1 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wnf 1706  wcel 1988  {crab 2913  Vcvv 3195  [wsbc 3429  cop 4174  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153
This theorem is referenced by:  mpt2xopovel  7331  mpt2xopoveqd  7332
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