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Theorem mpoxopoveq 6241
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 9 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑}))
3 fveq2 5516 . . . . 5 (𝑥 = ⟨𝑉, 𝑊⟩ → (1st𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
4 op1stg 6151 . . . . . 6 ((𝑉𝑋𝑊𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
54adantr 276 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
63, 5sylan9eqr 2232 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ 𝑥 = ⟨𝑉, 𝑊⟩) → (1st𝑥) = 𝑉)
76adantrr 479 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (1st𝑥) = 𝑉)
8 sbceq1a 2973 . . . . . 6 (𝑦 = 𝐾 → (𝜑[𝐾 / 𝑦]𝜑))
98adantl 277 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → (𝜑[𝐾 / 𝑦]𝜑))
109adantl 277 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝐾 / 𝑦]𝜑))
11 sbceq1a 2973 . . . . . 6 (𝑥 = ⟨𝑉, 𝑊⟩ → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1211adantr 276 . . . . 5 ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1312adantl 277 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → ([𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
1410, 13bitrd 188 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
157, 14rabeqbidv 2733 . 2 ((((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → {𝑛 ∈ (1st𝑥) ∣ 𝜑} = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
16 opexg 4229 . . 3 ((𝑉𝑋𝑊𝑌) → ⟨𝑉, 𝑊⟩ ∈ V)
1716adantr 276 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → ⟨𝑉, 𝑊⟩ ∈ V)
18 simpr 110 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → 𝐾𝑉)
19 rabexg 4147 . . 3 (𝑉𝑋 → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
2019ad2antrr 488 . 2 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
21 equid 1701 . . 3 𝑧 = 𝑧
22 nfvd 1529 . . 3 (𝑧 = 𝑧 → Ⅎ𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2321, 22ax-mp 5 . 2 𝑥((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
24 nfvd 1529 . . 3 (𝑧 = 𝑧 → Ⅎ𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉))
2521, 24ax-mp 5 . 2 𝑦((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉)
26 nfcv 2319 . 2 𝑦𝑉, 𝑊
27 nfcv 2319 . 2 𝑥𝐾
28 nfsbc1v 2982 . . 3 𝑥[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
29 nfcv 2319 . . 3 𝑥𝑉
3028, 29nfrabxy 2658 . 2 𝑥{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
31 nfsbc1v 2982 . . . 4 𝑦[𝐾 / 𝑦]𝜑
3226, 31nfsbc 2984 . . 3 𝑦[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
33 nfcv 2319 . . 3 𝑦𝑉
3432, 33nfrabxy 2658 . 2 𝑦{𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpodxf 6000 1 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wnf 1460  wcel 2148  {crab 2459  Vcvv 2738  [wsbc 2963  cop 3596  cfv 5217  (class class class)co 5875  cmpo 5877  1st c1st 6139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141
This theorem is referenced by:  mpoxopovel  6242
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