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Theorem sprmpod 8159
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 7370 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸))
43breqd 5120 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑥(𝑣𝑅𝑒)𝑦𝑥(𝑉𝑅𝐸)𝑦))
54adantl 483 . . . 4 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑥(𝑣𝑅𝑒)𝑦𝑥(𝑉𝑅𝐸)𝑦))
6 sprmpod.2 . . . . 5 ((𝜑𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))
763expb 1121 . . . 4 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝜒𝜓))
85, 7anbi12d 632 . . 3 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦𝜓)))
98opabbidv 5175 . 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 496 . 2 (𝜑𝑉 ∈ V)
1210simprd 497 . 2 (𝜑𝐸 ∈ V)
13 sprmpod.4 . . 3 (𝜑 → ∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃))
14 sprmpod.5 . . 3 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)
15 opabbrex 7412 . . 3 ((∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
1613, 14, 15syl2anc 585 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
172, 9, 11, 12, 16ovmpod 7511 1 (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wcel 2107  Vcvv 3447   class class class wbr 5109  {copab 5171  (class class class)co 7361  cmpo 7363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366
This theorem is referenced by: (None)
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