Theorem sprmpod 8228

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.) (Revised by AV, 20-Jun-2019.)

Hypotheses

Ref	Expression
sprmpod.1	⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)})
sprmpod.2	⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓))
sprmpod.3	⊢ (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
sprmpod.4	⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃))
sprmpod.5	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)

Assertion

Ref	Expression
sprmpod	⊢ (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)})

Distinct variable groups:   𝑒,𝐸,𝑣,𝑥,𝑦   𝑅,𝑒,𝑣   𝑒,𝑉,𝑣,𝑥,𝑦   𝜑,𝑒,𝑣,𝑥,𝑦   𝜓,𝑒,𝑣

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑣,𝑒)   𝜃(𝑥,𝑦,𝑣,𝑒)   𝑅(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑣,𝑒)

Proof of Theorem sprmpod

Step	Hyp	Ref	Expression
1		sprmpod.1	. . 3 ⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)})
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)}))
3		oveq12 7419	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸))
4	3	breqd 5135	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑥(𝑣𝑅𝑒)𝑦 ↔ 𝑥(𝑉𝑅𝐸)𝑦))
5	4	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑥(𝑣𝑅𝑒)𝑦 ↔ 𝑥(𝑉𝑅𝐸)𝑦))
6		sprmpod.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓))
7	6	3expb 1120	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝜒 ↔ 𝜓))
8	5, 7	anbi12d 632	. . 3 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)))
9	8	opabbidv 5190	. 2 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)})
10		sprmpod.3	. . 3 ⊢ (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
11	10	simpld 494	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
12	10	simprd 495	. 2 ⊢ (𝜑 → 𝐸 ∈ V)
13		sprmpod.4	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃))
14		sprmpod.5	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)
15		opabbrex 7463	. . 3 ⊢ ((∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V)
16	13, 14, 15	syl2anc 584	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V)
17	2, 9, 11, 12, 16	ovmpod 7564	1 ⊢ (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  Vcvv 3464   class class class wbr 5124  {copab 5186  (class class class)co 7410   ∈ cmpo 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415

This theorem is referenced by: (None)