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Theorem sprmpt2 5767
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 5433 . . . . . 6 ((v = 𝑉 𝑒 = 𝐸) → (v𝑊𝑒) = (𝑉𝑊𝐸))
43adantl 262 . . . . 5 (((𝑉 V 𝐸 V) (v = 𝑉 𝑒 = 𝐸)) → (v𝑊𝑒) = (𝑉𝑊𝐸))
54breqd 3738 . . . 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 3786 . 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 5460 . 2 ((𝑉 V 𝐸 V) → {⟨f, 𝑝⟩ ∣ (f(𝑉𝑊𝐸)𝑝 ψ)} V)
152, 9, 10, 11, 14ovmpt2d 5539 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 1223   wcel 1366  Vcvv 2526   class class class wbr 3727  {copab 3780  (class class class)co 5424  cmpt2 5426
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907  ax-setind 4192
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-fal 1229  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ne 2179  df-ral 2280  df-rex 2281  df-v 2528  df-sbc 2733  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-opab 3782  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-iota 4782  df-fun 4819  df-fv 4825  df-ov 5427  df-oprab 5428  df-mpt2 5429
This theorem is referenced by: (None)
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