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Theorem mpoxopoveqd 8223
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 8221 . . . 4 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
32ex 411 . . 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 3037 . . . . . 6 (𝐾𝑉 ↔ ¬ 𝐾𝑉)
71mpoxopynvov0 8220 . . . . . 6 (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
86, 7sylbir 234 . . . . 5 𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
98adantr 479 . . . 4 ((¬ 𝐾𝑉𝜓) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)
10 mpoxopoveqd.2 . . . . . 6 ((𝜓 ∧ ¬ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)
1110eqcomd 2731 . . . . 5 ((𝜓 ∧ ¬ 𝐾𝑉) → ∅ = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
1211ancoms 457 . . . 4 ((¬ 𝐾𝑉𝜓) → ∅ = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
139, 12eqtrd 2765 . . 3 ((¬ 𝐾𝑉𝜓) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
1413ex 411 . 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 394   = wceq 1533  wcel 2098  wnel 3036  {crab 3419  Vcvv 3463  [wsbc 3769  c0 4318  cop 4630  cfv 6542  (class class class)co 7415  cmpo 7417  1st c1st 7987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990
This theorem is referenced by: (None)
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