Theorem sprmpt2 5857

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 5521	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
4	3	adantl 262	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑣𝑊𝑒) = (𝑉𝑊𝐸))
5	4	breqd 3775	. . . 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 262	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝜒 ↔ 𝜓))
8	5, 7	anbi12d 442	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑓(𝑣𝑊𝑒)𝑝 ∧ 𝜒) ↔ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)))
9	8	opabbidv 3823	. 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)𝑝 → 𝜃))
13		sprmpt2.4	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)
14	12, 13	opabbrex 5549	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ V)
15	2, 9, 10, 11, 14	ovmpt2d 5628	1 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑀𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557   class class class wbr 3764  {copab 3817  (class class class)co 5512   ↦ cmpt2 5514

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517

This theorem is referenced by: (None)