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Theorem sprmpt2 5888
 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 5549 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
43adantl 266 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
54breqd 3803 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑓(𝑣𝑊𝑒)𝑝𝑓(𝑉𝑊𝐸)𝑝))
6 sprmpt2.2 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))
76adantl 266 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝜒𝜓))
85, 7anbi12d 450 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((𝑓(𝑣𝑊𝑒)𝑝𝜒) ↔ (𝑓(𝑉𝑊𝐸)𝑝𝜓)))
98opabbidv 3851 . 2 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝𝜒)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)})
10 simpl 106 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
11 simpr 107 . 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 5577 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ∈ V)
152, 9, 10, 11, 14ovmpt2d 5656 1 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑀𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  Vcvv 2574   class class class wbr 3792  {copab 3845  (class class class)co 5540   ↦ cmpt2 5542 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545 This theorem is referenced by: (None)
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