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Theorem mpt2xopoveq 5910
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 9 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑}))
3 fveq2 5230 . . . . 5 (𝑥 = ⟨𝑉, 𝑊⟩ → (1st𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
4 op1stg 5829 . . . . . 6 ((𝑉𝑋𝑊𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
54adantr 270 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
63, 5sylan9eqr 2137 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ 𝑥 = ⟨𝑉, 𝑊⟩) → (1st𝑥) = 𝑉)
76adantrr 463 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (1st𝑥) = 𝑉)
8 sbceq1a 2833 . . . . . 6 (𝑦 = 𝐾 → (𝜑[𝐾 / 𝑦]𝜑))
98adantl 271 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → (𝜑[𝐾 / 𝑦]𝜑))
109adantl 271 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝐾 / 𝑦]𝜑))
11 sbceq1a 2833 . . . . . 6 (𝑥 = ⟨𝑉, 𝑊⟩ → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1211adantr 270 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1312adantl 271 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1410, 13bitrd 186 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
157, 14rabeqbidv 2605 . 2 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → {𝑛 ∈ (1st𝑥) ∣ 𝜑} = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
16 opexg 4011 . . 3 ((𝑉𝑋𝑊𝑌) → ⟨𝑉, 𝑊⟩ ∈ V)
1716adantr 270 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → ⟨𝑉, 𝑊⟩ ∈ V)
18 simpr 108 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → 𝐾𝑉)
19 rabexg 3941 . . 3 (𝑉𝑋 → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
2019ad2antrr 472 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
21 equid 1630 . . 3 𝑧 = 𝑧
22 nfvd 1463 . . 3 (𝑧 = 𝑧 → Ⅎ𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2321, 22ax-mp 7 . 2 𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
24 nfvd 1463 . . 3 (𝑧 = 𝑧 → Ⅎ𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2521, 24ax-mp 7 . 2 𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
26 nfcv 2223 . 2 𝑦𝑉, 𝑊
27 nfcv 2223 . 2 𝑥𝐾
28 nfsbc1v 2842 . . 3 𝑥[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
29 nfcv 2223 . . 3 𝑥𝑉
3028, 29nfrabxy 2539 . 2 𝑥{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
31 nfsbc1v 2842 . . . 4 𝑦[𝐾 / 𝑦]𝜑
3226, 31nfsbc 2844 . . 3 𝑦[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
33 nfcv 2223 . . 3 𝑦𝑉
3432, 33nfrabxy 2539 . 2 𝑦{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 5678 1 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wnf 1390  wcel 1434  {crab 2357  Vcvv 2610  [wsbc 2824  cop 3419  cfv 4952  (class class class)co 5564  cmpt2 5566  1st c1st 5817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819
This theorem is referenced by:  mpt2xopovel  5911
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