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Theorem sprmpod 8234
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 7433 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸))
43breqd 5161 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑥(𝑣𝑅𝑒)𝑦𝑥(𝑉𝑅𝐸)𝑦))
54adantl 480 . . . 4 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑥(𝑣𝑅𝑒)𝑦𝑥(𝑉𝑅𝐸)𝑦))
6 sprmpod.2 . . . . 5 ((𝜑𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))
763expb 1117 . . . 4 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝜒𝜓))
85, 7anbi12d 630 . . 3 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦𝜓)))
98opabbidv 5216 . 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 493 . 2 (𝜑𝑉 ∈ V)
1210simprd 494 . 2 (𝜑𝐸 ∈ V)
13 sprmpod.4 . . 3 (𝜑 → ∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃))
14 sprmpod.5 . . 3 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)
15 opabbrex 7475 . . 3 ((∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
1613, 14, 15syl2anc 582 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
172, 9, 11, 12, 16ovmpod 7577 1 (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wal 1531   = wceq 1533  wcel 2098  Vcvv 3471   class class class wbr 5150  {copab 5212  (class class class)co 7424  cmpo 7426
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 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429
This theorem is referenced by: (None)
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