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Theorem relopabVD 43662
Description: Virtual deduction proof of relopab 5825. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. relopab 5825 is relopabVD 43662 without virtual deductions and was automatically derived from relopabVD 43662.
1:: (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   )
2:1: (   𝑦 = 𝑣   ▶   𝑥   ,   𝑦⟩ = ⟨𝑥   ,   𝑣    )
3:: (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
4:3: (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥   ,   𝑣⟩ = ⟨ 𝑢, 𝑣   )
5:2,4: (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥   ,   𝑦⟩ = ⟨ 𝑢, 𝑣   )
6:5: (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥   ,   𝑦 ⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   )
7:6: (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,    𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   )
8:7: (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦 ⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
9:8: (∃𝑣𝑦 = 𝑣 → ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
90:: (𝑣 = 𝑦𝑦 = 𝑣)
91:90: (∃𝑣𝑣 = 𝑦 ↔ ∃𝑣𝑦 = 𝑣)
92:: 𝑣𝑣 = 𝑦
10:91,92: 𝑣𝑦 = 𝑣
11:9,10: 𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
12:11: (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
13:: (∃𝑣(𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢 , 𝑣⟩) → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩))
14:12,13: (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
15:14: (∃𝑢𝑥 = 𝑢 → ∃𝑢(𝑧 = ⟨𝑥   ,   𝑦 ⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩))
150:: (𝑢 = 𝑥𝑥 = 𝑢)
151:150: (∃𝑢𝑢 = 𝑥 ↔ ∃𝑢𝑥 = 𝑢)
152:: 𝑢𝑢 = 𝑥
16:151,152: 𝑢𝑥 = 𝑢
17:15,16: 𝑢(𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑣𝑧 = ⟨ 𝑢, 𝑣⟩)
18:17: (𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢𝑣𝑧 = ⟨ 𝑢, 𝑣⟩)
19:18: (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑦𝑢 𝑣𝑧 = ⟨𝑢, 𝑣⟩)
20:: (∃𝑦𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → 𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
21:19,20: (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
22:21: (∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑥 𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
23:: (∃𝑥𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → 𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
24:22,23: (∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢 𝑣𝑧 = ⟨𝑢, 𝑣⟩)
25:24: {𝑧 ∣ ∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩}
26:: 𝑥 ∈ V
27:: 𝑦 ∈ V
28:26,27: (𝑥 ∈ V ∧ 𝑦 ∈ V)
29:28: (𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ (𝑧 = ⟨𝑥   ,   𝑦 ⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
30:29: (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑦(𝑧 = 𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
31:30: (∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑥 𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
32:31: {𝑧 ∣ ∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} = { 𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
320:25,32: {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩}
33:: 𝑢 ∈ V
34:: 𝑣 ∈ V
35:33,34: (𝑢 ∈ V ∧ 𝑣 ∈ V)
36:35: (𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ (𝑧 = ⟨𝑢   ,   𝑣 ⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
37:36: (∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑣(𝑧 = 𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
38:37: (∃𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑢 𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
39:38: {𝑧 ∣ ∃𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩} = { 𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
40:320,39: {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
41:: {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) }
42:: {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) }
43:40,41,42: {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
44:: {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = (V × V)
45:43,44: {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} ⊆ (V × V)
46:28: (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
47:46: {⟨𝑥   ,   𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥   ,   𝑦 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
48:45,47: {⟨𝑥   ,   𝑦⟩ ∣ 𝜑} ⊆ (V × V)
qed:48: Rel {⟨𝑥   ,   𝑦⟩ ∣ 𝜑}
(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
relopabVD Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem relopabVD
Dummy variables 𝑧 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3479 . . . . . 6 𝑥 ∈ V
2 vex 3479 . . . . . 6 𝑦 ∈ V
31, 2pm3.2i 472 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . . 4 (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 5551 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
63biantru 531 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
76exbii 1851 . . . . . . . . 9 (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
87exbii 1851 . . . . . . . 8 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
98abbii 2803 . . . . . . 7 {𝑧 ∣ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
10 ax6ev 1974 . . . . . . . . . . . . . . 15 𝑢 𝑢 = 𝑥
11 equcom 2022 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥𝑥 = 𝑢)
1211exbii 1851 . . . . . . . . . . . . . . 15 (∃𝑢 𝑢 = 𝑥 ↔ ∃𝑢 𝑥 = 𝑢)
1310, 12mpbi 229 . . . . . . . . . . . . . 14 𝑢 𝑥 = 𝑢
14 ax6ev 1974 . . . . . . . . . . . . . . . . . . 19 𝑣 𝑣 = 𝑦
15 equcom 2022 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑦𝑦 = 𝑣)
1615exbii 1851 . . . . . . . . . . . . . . . . . . 19 (∃𝑣 𝑣 = 𝑦 ↔ ∃𝑣 𝑦 = 𝑣)
1714, 16mpbi 229 . . . . . . . . . . . . . . . . . 18 𝑣 𝑦 = 𝑣
18 idn1 43335 . . . . . . . . . . . . . . . . . . . . . . . 24 (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   )
19 opeq2 4875 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
2018, 19e1a 43388 . . . . . . . . . . . . . . . . . . . . . . 23 (   𝑦 = 𝑣   ▶   𝑥, 𝑦⟩ = ⟨𝑥, 𝑣   )
21 idn2 43374 . . . . . . . . . . . . . . . . . . . . . . . 24 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
22 opeq1 4874 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
2321, 22e2 43392 . . . . . . . . . . . . . . . . . . . . . . 23 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥, 𝑣⟩ = ⟨𝑢, 𝑣   )
24 eqeq1 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩))
2524biimprd 247 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩))
2620, 23, 25e12 43485 . . . . . . . . . . . . . . . . . . . . . 22 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥, 𝑦⟩ = ⟨𝑢, 𝑣   )
27 eqeq2 2745 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑢, 𝑣⟩))
2827biimpd 228 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
2926, 28e2 43392 . . . . . . . . . . . . . . . . . . . . 21 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   )
3029in2 43366 . . . . . . . . . . . . . . . . . . . 20 (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   )
3130in1 43332 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
3231eximi 1838 . . . . . . . . . . . . . . . . . 18 (∃𝑣 𝑦 = 𝑣 → ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
3317, 32ax-mp 5 . . . . . . . . . . . . . . . . 17 𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
343319.37iv 1953 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
35 19.37v 1996 . . . . . . . . . . . . . . . . 17 (∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
3635biimpi 215 . . . . . . . . . . . . . . . 16 (∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
3734, 36syl 17 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
3837eximi 1838 . . . . . . . . . . . . . 14 (∃𝑢 𝑥 = 𝑢 → ∃𝑢(𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
3913, 38ax-mp 5 . . . . . . . . . . . . 13 𝑢(𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
403919.37iv 1953 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4140eximi 1838 . . . . . . . . . . 11 (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
42 19.9v 1988 . . . . . . . . . . . 12 (∃𝑦𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4342biimpi 215 . . . . . . . . . . 11 (∃𝑦𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4441, 43syl 17 . . . . . . . . . 10 (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4544eximi 1838 . . . . . . . . 9 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
46 19.9v 1988 . . . . . . . . . 10 (∃𝑥𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4746biimpi 215 . . . . . . . . 9 (∃𝑥𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4845, 47syl 17 . . . . . . . 8 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4948ss2abi 4064 . . . . . . 7 {𝑧 ∣ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
509, 49eqsstrri 4018 . . . . . 6 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
51 vex 3479 . . . . . . . . . . 11 𝑢 ∈ V
52 vex 3479 . . . . . . . . . . 11 𝑣 ∈ V
5351, 52pm3.2i 472 . . . . . . . . . 10 (𝑢 ∈ V ∧ 𝑣 ∈ V)
5453biantru 531 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ ↔ (𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
5554exbii 1851 . . . . . . . 8 (∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
5655exbii 1851 . . . . . . 7 (∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
5756abbii 2803 . . . . . 6 {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩} = {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
5850, 57sseqtri 4019 . . . . 5 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
59 df-opab 5212 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
60 df-opab 5212 . . . . 5 {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
6158, 59, 603sstr4i 4026 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
62 df-xp 5683 . . . . 5 (V × V) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
6362eqcomi 2742 . . . 4 {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = (V × V)
6461, 63sseqtri 4019 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ (V × V)
655, 64sstri 3992 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
66 df-rel 5684 . . 3 (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V))
6766biimpri 227 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V) → Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6865, 67e0a 43533 1 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  Vcvv 3475  wss 3949  cop 4635  {copab 5211   × cxp 5675  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684  df-vd1 43331  df-vd2 43339
This theorem is referenced by: (None)
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