Theorem sprmpt2 6021

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 5675	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
4	3	adantl 272	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
5	4	breqd 3862	. . . 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 272	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝜒 ↔ 𝜓))
8	5, 7	anbi12d 458	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑓(𝑣𝑊𝑒)𝑝 ∧ 𝜒) ↔ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)))
9	8	opabbidv 3910	. 2 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝 ∧ 𝜒)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)})
10		simpl 108	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
11		simpr 109	. 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 5707	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ V)
15	2, 9, 10, 11, 14	ovmpt2d 5786	1 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑀𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290   ∈ wcel 1439  Vcvv 2620   class class class wbr 3851  {copab 3904  (class class class)co 5666   ↦ cmpt2 5668

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-setind 4366

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671

This theorem is referenced by: (None)