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Theorem sprmpt2 5798
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)𝑝 χ)})
sprmpt2.2 ((v = 𝑉 𝑒 = 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)𝑝 ψ)})
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)𝑝 χ)})
21a1i 9 . 2 ((𝑉 V 𝐸 V) → 𝑀 = (v V, 𝑒 V ↦ {⟨f, 𝑝⟩ ∣ (f(v𝑊𝑒)𝑝 χ)}))
3 oveq12 5464 . . . . . 6 ((v = 𝑉 𝑒 = 𝐸) → (v𝑊𝑒) = (𝑉𝑊𝐸))
43adantl 262 . . . . 5 (((𝑉 V 𝐸 V) (v = 𝑉 𝑒 = 𝐸)) → (v𝑊𝑒) = (𝑉𝑊𝐸))
54breqd 3766 . . . 4 (((𝑉 V 𝐸 V) (v = 𝑉 𝑒 = 𝐸)) → (f(v𝑊𝑒)𝑝f(𝑉𝑊𝐸)𝑝))
6 sprmpt2.2 . . . . 5 ((v = 𝑉 𝑒 = 𝐸) → (χψ))
76adantl 262 . . . 4 (((𝑉 V 𝐸 V) (v = 𝑉 𝑒 = 𝐸)) → (χψ))
85, 7anbi12d 442 . . 3 (((𝑉 V 𝐸 V) (v = 𝑉 𝑒 = 𝐸)) → ((f(v𝑊𝑒)𝑝 χ) ↔ (f(𝑉𝑊𝐸)𝑝 ψ)))
98opabbidv 3814 . 2 (((𝑉 V 𝐸 V) (v = 𝑉 𝑒 = 𝐸)) → {⟨f, 𝑝⟩ ∣ (f(v𝑊𝑒)𝑝 χ)} = {⟨f, 𝑝⟩ ∣ (f(𝑉𝑊𝐸)𝑝 ψ)})
10 simpl 102 . 2 ((𝑉 V 𝐸 V) → 𝑉 V)
11 simpr 103 . 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 5491 . 2 ((𝑉 V 𝐸 V) → {⟨f, 𝑝⟩ ∣ (f(𝑉𝑊𝐸)𝑝 ψ)} V)
152, 9, 10, 11, 14ovmpt2d 5570 1 ((𝑉 V 𝐸 V) → (𝑉𝑀𝐸) = {⟨f, 𝑝⟩ ∣ (f(𝑉𝑊𝐸)𝑝 ψ)})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551   class class class wbr 3755  {copab 3808  (class class class)co 5455  cmpt2 5457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460
This theorem is referenced by: (None)
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