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Theorem sprmpt2 5989
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.)
Hypotheses
Ref Expression
sprmpt2.1 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝𝜒)})
sprmpt2.2 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))
sprmpt2.3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝𝜃))
sprmpt2.4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)
Assertion
Ref Expression
sprmpt2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑀𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)})
Distinct variable groups:   𝑒,𝐸,𝑓,𝑝,𝑣   𝑒,𝑉,𝑓,𝑝,𝑣   𝑒,𝑊,𝑣   𝜓,𝑒,𝑣
Allowed substitution hints:   𝜓(𝑓,𝑝)   𝜒(𝑣,𝑒,𝑓,𝑝)   𝜃(𝑣,𝑒,𝑓,𝑝)   𝑀(𝑣,𝑒,𝑓,𝑝)   𝑊(𝑓,𝑝)

Proof of Theorem sprmpt2
StepHypRef Expression
1 sprmpt2.1 . . 3 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝𝜒)})
21a1i 9 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝𝜒)}))
3 oveq12 5643 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
43adantl 271 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
54breqd 3848 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑓(𝑣𝑊𝑒)𝑝𝑓(𝑉𝑊𝐸)𝑝))
6 sprmpt2.2 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))
76adantl 271 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝜒𝜓))
85, 7anbi12d 457 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((𝑓(𝑣𝑊𝑒)𝑝𝜒) ↔ (𝑓(𝑉𝑊𝐸)𝑝𝜓)))
98opabbidv 3896 . 2 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝𝜒)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)})
10 simpl 107 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
11 simpr 108 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V)
12 sprmpt2.3 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝𝜃))
13 sprmpt2.4 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)
1412, 13opabbrex 5675 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ∈ V)
152, 9, 10, 11, 14ovmpt2d 5754 1 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑀𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  Vcvv 2619   class class class wbr 3837  {copab 3890  (class class class)co 5634  cmpt2 5636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639
This theorem is referenced by: (None)
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