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Theorem mpt2xopoveq 5885
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 5205 . . . . 5 (𝑥 = ⟨𝑉, 𝑊⟩ → (1st𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
4 op1stg 5804 . . . . . 6 ((𝑉𝑋𝑊𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
54adantr 265 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
63, 5sylan9eqr 2110 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ 𝑥 = ⟨𝑉, 𝑊⟩) → (1st𝑥) = 𝑉)
76adantrr 456 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (1st𝑥) = 𝑉)
8 sbceq1a 2795 . . . . . 6 (𝑦 = 𝐾 → (𝜑[𝐾 / 𝑦]𝜑))
98adantl 266 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → (𝜑[𝐾 / 𝑦]𝜑))
109adantl 266 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝐾 / 𝑦]𝜑))
11 sbceq1a 2795 . . . . . 6 (𝑥 = ⟨𝑉, 𝑊⟩ → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1211adantr 265 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1312adantl 266 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1410, 13bitrd 181 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
157, 14rabeqbidv 2569 . 2 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → {𝑛 ∈ (1st𝑥) ∣ 𝜑} = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
16 opexg 3991 . . 3 ((𝑉𝑋𝑊𝑌) → ⟨𝑉, 𝑊⟩ ∈ V)
1716adantr 265 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → ⟨𝑉, 𝑊⟩ ∈ V)
18 simpr 107 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → 𝐾𝑉)
19 rabexg 3927 . . 3 (𝑉𝑋 → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
2019ad2antrr 465 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
21 equid 1605 . . 3 𝑧 = 𝑧
22 nfvd 1438 . . 3 (𝑧 = 𝑧 → Ⅎ𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2321, 22ax-mp 7 . 2 𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
24 nfvd 1438 . . 3 (𝑧 = 𝑧 → Ⅎ𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2521, 24ax-mp 7 . 2 𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
26 nfcv 2194 . 2 𝑦𝑉, 𝑊
27 nfcv 2194 . 2 𝑥𝐾
28 nfsbc1v 2804 . . 3 𝑥[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
29 nfcv 2194 . . 3 𝑥𝑉
3028, 29nfrabxy 2507 . 2 𝑥{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
31 nfsbc1v 2804 . . . 4 𝑦[𝐾 / 𝑦]𝜑
3226, 31nfsbc 2806 . . 3 𝑦[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
33 nfcv 2194 . . 3 𝑦𝑉
3432, 33nfrabxy 2507 . 2 𝑦{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 5653 1 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnf 1365  wcel 1409  {crab 2327  Vcvv 2574  [wsbc 2786  cop 3405  cfv 4929  (class class class)co 5539  cmpt2 5541  1st c1st 5792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794
This theorem is referenced by:  mpt2xopovel  5886
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