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Theorem sprmpod 8207
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 7413 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸))
43breqd 5152 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑥(𝑣𝑅𝑒)𝑦𝑥(𝑉𝑅𝐸)𝑦))
54adantl 481 . . . 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 5207 . 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 494 . 2 (𝜑𝑉 ∈ V)
1210simprd 495 . 2 (𝜑𝐸 ∈ V)
13 sprmpod.4 . . 3 (𝜑 → ∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃))
14 sprmpod.5 . . 3 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)
15 opabbrex 7455 . . 3 ((∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
1613, 14, 15syl2anc 583 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
172, 9, 11, 12, 16ovmpod 7555 1 (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084  wal 1531   = wceq 1533  wcel 2098  Vcvv 3468   class class class wbr 5141  {copab 5203  (class class class)co 7404  cmpo 7406
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409
This theorem is referenced by: (None)
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