Theorem sprmpt2 5880

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) → (𝜒 ↔ 𝜓))
sprmpt2.3	⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝 → 𝜃))
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

Step	Hyp	Ref	Expression
1		sprmpt2.1	. . 3 ⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝 ∧ 𝜒)})
2	1	a1i 9	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝 ∧ 𝜒)}))
3		oveq12 5541	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
4	3	adantl 271	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
5	4	breqd 3796	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑓(𝑣𝑊𝑒)𝑝 ↔ 𝑓(𝑉𝑊𝐸)𝑝))
6		sprmpt2.2	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓))
7	6	adantl 271	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝜒 ↔ 𝜓))
8	5, 7	anbi12d 456	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑓(𝑣𝑊𝑒)𝑝 ∧ 𝜒) ↔ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)))
9	8	opabbidv 3844	. 2 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝 ∧ 𝜒)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)})
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)𝑝 → 𝜃))
13		sprmpt2.4	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)
14	12, 13	opabbrex 5569	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ V)
15	2, 9, 10, 11, 14	ovmpt2d 5648	1 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑀𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)})

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  Vcvv 2601   class class class wbr 3785  {copab 3838  (class class class)co 5532   ↦ cmpt2 5534

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537

This theorem is referenced by: (None)