Theorem sprmpod 8225

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 7421	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸))
4	3	breqd 5114	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑥(𝑣𝑅𝑒)𝑦 ↔ 𝑥(𝑉𝑅𝐸)𝑦))
5	4	adantl 487	. . . 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 1138	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝜒 ↔ 𝜓))
8	5, 7	anbi12d 644	. . 3 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)))
9	8	opabbidv 5171	. 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 500	. 2 ⊢ (𝜑 → 𝑉 ∈ V)
12	10	simprd 501	. 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 7465	. . 3 ⊢ ((∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V)
16	13, 14, 15	syl2anc 596	. 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)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  Vcvv 3451   class class class wbr 5103  {copab 5167  (class class class)co 7412   ∈ cmpo 7414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417

This theorem is used by: (None)