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Theorem sprmpt2d 7511
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
sprmpt2d.1 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦𝜒)})
sprmpt2d.2 ((𝜑𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))
sprmpt2d.3 (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
sprmpt2d.4 (𝜑 → ∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃))
sprmpt2d.5 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)
Assertion
Ref Expression
sprmpt2d (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 sprmpt2d
StepHypRef Expression
1 sprmpt2d.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 6814 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸))
43breqd 4807 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑥(𝑣𝑅𝑒)𝑦𝑥(𝑉𝑅𝐸)𝑦))
54adantl 473 . . . 4 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑥(𝑣𝑅𝑒)𝑦𝑥(𝑉𝑅𝐸)𝑦))
6 sprmpt2d.2 . . . . 5 ((𝜑𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))
763expb 1113 . . . 4 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝜒𝜓))
85, 7anbi12d 749 . . 3 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦𝜓)))
98opabbidv 4860 . 2 ((𝜑 ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})
10 sprmpt2d.3 . . 3 (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
1110simpld 477 . 2 (𝜑𝑉 ∈ V)
1210simprd 482 . 2 (𝜑𝐸 ∈ V)
13 sprmpt2d.4 . . 3 (𝜑 → ∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃))
14 sprmpt2d.5 . . 3 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)
15 opabbrex 6852 . . 3 ((∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
1613, 14, 15syl2anc 696 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)} ∈ V)
172, 9, 11, 12, 16ovmpt2d 6945 1 (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wal 1622   = wceq 1624  wcel 2131  Vcvv 3332   class class class wbr 4796  {copab 4856  (class class class)co 6805  cmpt2 6807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810
This theorem is referenced by: (None)
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