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Theorem sprmpod 8147
Description: The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 20-Jun-2019.)
Hypotheses
Ref Expression
sprmpod.1 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦𝜒)})
sprmpod.2 ((𝜑𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))
sprmpod.3 (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
sprmpod.4 (𝜑 → ∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃))
sprmpod.5 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)
Assertion
Ref Expression
sprmpod (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})
Distinct variable groups:   𝑒,𝐸,𝑣,𝑥,𝑦   𝑅,𝑒,𝑣   𝑒,𝑉,𝑣,𝑥,𝑦   𝜑,𝑒,𝑣,𝑥,𝑦   𝜓,𝑒,𝑣
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑣,𝑒)   𝜃(𝑥,𝑦,𝑣,𝑒)   𝑅(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑣,𝑒)

Proof of Theorem sprmpod
StepHypRef Expression
1 sprmpod.1 . . 3 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦𝜒)})
21a1i 11 . 2 (𝜑𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦𝜒)}))
3 oveq12 7360 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸))
43breqd 5114 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑥(𝑣𝑅𝑒)𝑦𝑥(𝑉𝑅𝐸)𝑦))
54adantl 482 . . . 4 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑥(𝑣𝑅𝑒)𝑦𝑥(𝑉𝑅𝐸)𝑦))
6 sprmpod.2 . . . . 5 ((𝜑𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))
763expb 1120 . . . 4 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝜒𝜓))
85, 7anbi12d 631 . . 3 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦𝜓)))
98opabbidv 5169 . 2 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})
10 sprmpod.3 . . 3 (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
1110simpld 495 . 2 (𝜑𝑉 ∈ V)
1210simprd 496 . 2 (𝜑𝐸 ∈ V)
13 sprmpod.4 . . 3 (𝜑 → ∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃))
14 sprmpod.5 . . 3 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)
15 opabbrex 7402 . . 3 ((∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
1613, 14, 15syl2anc 584 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
172, 9, 11, 12, 16ovmpod 7501 1 (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  Vcvv 3443   class class class wbr 5103  {copab 5165  (class class class)co 7351  cmpo 7353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356
This theorem is referenced by: (None)
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