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Theorem mpt2xopoveqd 7234
 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, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
Hypotheses
Ref Expression
mpt2xopoveq.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
mpt2xopoveqd.1 (𝜓 → (𝑉𝑋𝑊𝑌))
mpt2xopoveqd.2 ((𝜓 ∧ ¬ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)
Assertion
Ref Expression
mpt2xopoveqd (𝜓 → (⟨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   𝑥,𝐹
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑛)   𝜓(𝑥,𝑦,𝑛)   𝐹(𝑦,𝑛)

Proof of Theorem mpt2xopoveqd
StepHypRef Expression
1 mpt2xopoveqd.1 . . . 4 (𝜓 → (𝑉𝑋𝑊𝑌))
2 mpt2xopoveq.f . . . . . 6 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
32mpt2xopoveq 7232 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
43ex 449 . . . 4 ((𝑉𝑋𝑊𝑌) → (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
51, 4syl 17 . . 3 (𝜓 → (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
65com12 32 . 2 (𝐾𝑉 → (𝜓 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
7 df-nel 2783 . . . . . 6 (𝐾𝑉 ↔ ¬ 𝐾𝑉)
82mpt2xopynvov0 7231 . . . . . 6 (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
97, 8sylbir 224 . . . . 5 𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
109adantr 480 . . . 4 ((¬ 𝐾𝑉𝜓) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
11 mpt2xopoveqd.2 . . . . . 6 ((𝜓 ∧ ¬ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)
1211eqcomd 2616 . . . . 5 ((𝜓 ∧ ¬ 𝐾𝑉) → ∅ = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
1312ancoms 468 . . . 4 ((¬ 𝐾𝑉𝜓) → ∅ = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
1410, 13eqtrd 2644 . . 3 ((¬ 𝐾𝑉𝜓) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
1514ex 449 . 2 𝐾𝑉 → (𝜓 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
166, 15pm2.61i 175 1 (𝜓 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173  [wsbc 3402  ∅c0 3874  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060 This theorem is referenced by: (None)
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