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Theorem mpoxopoveqd 8212
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
mpoxopoveq.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
mpoxopoveqd.1 (𝜓 → (𝑉𝑋𝑊𝑌))
mpoxopoveqd.2 ((𝜓 ∧ ¬ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)
Assertion
Ref Expression
mpoxopoveqd (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 mpoxopoveqd
StepHypRef Expression
1 mpoxopoveq.f . . . . 5 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
21mpoxopoveq 8210 . . . 4 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
32ex 412 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
4 mpoxopoveqd.1 . . 3 (𝜓 → (𝑉𝑋𝑊𝑌))
53, 4syl11 33 . 2 (𝐾𝑉 → (𝜓 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
6 df-nel 3046 . . . . . 6 (𝐾𝑉 ↔ ¬ 𝐾𝑉)
71mpoxopynvov0 8209 . . . . . 6 (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
86, 7sylbir 234 . . . . 5 𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
98adantr 480 . . . 4 ((¬ 𝐾𝑉𝜓) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
10 mpoxopoveqd.2 . . . . . 6 ((𝜓 ∧ ¬ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)
1110eqcomd 2737 . . . . 5 ((𝜓 ∧ ¬ 𝐾𝑉) → ∅ = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
1211ancoms 458 . . . 4 ((¬ 𝐾𝑉𝜓) → ∅ = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
139, 12eqtrd 2771 . . 3 ((¬ 𝐾𝑉𝜓) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
1413ex 412 . 2 𝐾𝑉 → (𝜓 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
155, 14pm2.61i 182 1 (𝜓 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2105  wnel 3045  {crab 3431  Vcvv 3473  [wsbc 3777  c0 4322  cop 4634  cfv 6543  (class class class)co 7412  cmpo 7414  1st c1st 7977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980
This theorem is referenced by: (None)
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