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Theorem relop 5689
Description: A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) A relation is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a relation is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is relsnopg 5644, as funsng 6379 is to funop 6892. (New usage is discouraged.)
Hypotheses
Ref Expression
relop.1 𝐴 ∈ V
relop.2 𝐵 ∈ V
Assertion
Ref Expression
relop (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem relop
Dummy variables 𝑤 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rel 5530 . 2 (Rel ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ ⊆ (V × V))
2 dfss2 3904 . . . . 5 (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ ∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)))
3 relop.1 . . . . . . . . . 10 𝐴 ∈ V
4 relop.2 . . . . . . . . . 10 𝐵 ∈ V
53, 4elop 5327 . . . . . . . . 9 (𝑧 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}))
6 elvv 5594 . . . . . . . . 9 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
75, 6imbi12i 354 . . . . . . . 8 ((𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
8 jaob 959 . . . . . . . 8 (((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
97, 8bitri 278 . . . . . . 7 ((𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
109albii 1821 . . . . . 6 (∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ∀𝑧((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
11 19.26 1871 . . . . . 6 (∀𝑧((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
1210, 11bitri 278 . . . . 5 (∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
132, 12bitri 278 . . . 4 (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
14 snex 5300 . . . . . . 7 {𝐴} ∈ V
15 eqeq1 2805 . . . . . . . 8 (𝑧 = {𝐴} → (𝑧 = {𝐴} ↔ {𝐴} = {𝐴}))
16 eqeq1 2805 . . . . . . . . . 10 (𝑧 = {𝐴} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ {𝐴} = ⟨𝑥, 𝑦⟩))
17 eqcom 2808 . . . . . . . . . . 11 ({𝐴} = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = {𝐴})
18 vex 3447 . . . . . . . . . . . 12 𝑥 ∈ V
19 vex 3447 . . . . . . . . . . . 12 𝑦 ∈ V
2018, 19opeqsn 5362 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = {𝐴} ↔ (𝑥 = 𝑦𝐴 = {𝑥}))
2117, 20bitri 278 . . . . . . . . . 10 ({𝐴} = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑦𝐴 = {𝑥}))
2216, 21syl6bb 290 . . . . . . . . 9 (𝑧 = {𝐴} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑦𝐴 = {𝑥})))
23222exbidv 1925 . . . . . . . 8 (𝑧 = {𝐴} → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥})))
2415, 23imbi12d 348 . . . . . . 7 (𝑧 = {𝐴} → ((𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ({𝐴} = {𝐴} → ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥}))))
2514, 24spcv 3557 . . . . . 6 (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ({𝐴} = {𝐴} → ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥})))
26 sneq 4538 . . . . . . . . 9 (𝑤 = 𝑥 → {𝑤} = {𝑥})
2726eqeq2d 2812 . . . . . . . 8 (𝑤 = 𝑥 → (𝐴 = {𝑤} ↔ 𝐴 = {𝑥}))
2827cbvexvw 2044 . . . . . . 7 (∃𝑤 𝐴 = {𝑤} ↔ ∃𝑥 𝐴 = {𝑥})
29 ax6evr 2022 . . . . . . . . 9 𝑦 𝑥 = 𝑦
30 19.41v 1950 . . . . . . . . 9 (∃𝑦(𝑥 = 𝑦𝐴 = {𝑥}) ↔ (∃𝑦 𝑥 = 𝑦𝐴 = {𝑥}))
3129, 30mpbiran 708 . . . . . . . 8 (∃𝑦(𝑥 = 𝑦𝐴 = {𝑥}) ↔ 𝐴 = {𝑥})
3231exbii 1849 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥}) ↔ ∃𝑥 𝐴 = {𝑥})
33 eqid 2801 . . . . . . . 8 {𝐴} = {𝐴}
3433a1bi 366 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥}) ↔ ({𝐴} = {𝐴} → ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥})))
3528, 32, 343bitr2ri 303 . . . . . 6 (({𝐴} = {𝐴} → ∃𝑥𝑦(𝑥 = 𝑦𝐴 = {𝑥})) ↔ ∃𝑤 𝐴 = {𝑤})
3625, 35sylib 221 . . . . 5 (∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∃𝑤 𝐴 = {𝑤})
37 eqid 2801 . . . . . 6 {𝐴, 𝐵} = {𝐴, 𝐵}
38 prex 5301 . . . . . . 7 {𝐴, 𝐵} ∈ V
39 eqeq1 2805 . . . . . . . 8 (𝑧 = {𝐴, 𝐵} → (𝑧 = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴, 𝐵}))
40 eqeq1 2805 . . . . . . . . 9 (𝑧 = {𝐴, 𝐵} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ {𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
41402exbidv 1925 . . . . . . . 8 (𝑧 = {𝐴, 𝐵} → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
4239, 41imbi12d 348 . . . . . . 7 (𝑧 = {𝐴, 𝐵} → ((𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ({𝐴, 𝐵} = {𝐴, 𝐵} → ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩)))
4338, 42spcv 3557 . . . . . 6 (∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ({𝐴, 𝐵} = {𝐴, 𝐵} → ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
4437, 43mpi 20 . . . . 5 (∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩)
45 eqcom 2808 . . . . . . . . . 10 ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = {𝐴, 𝐵})
4618, 19, 3, 4opeqpr 5363 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ = {𝐴, 𝐵} ↔ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})))
4745, 46bitri 278 . . . . . . . . 9 ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ ↔ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})))
48 idd 24 . . . . . . . . . 10 (𝐴 = {𝑤} → ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
49 eqtr2 2822 . . . . . . . . . . . . . 14 ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → {𝑥, 𝑦} = {𝑤})
5018, 19preqsn 4755 . . . . . . . . . . . . . . 15 ({𝑥, 𝑦} = {𝑤} ↔ (𝑥 = 𝑦𝑦 = 𝑤))
5150simplbi 501 . . . . . . . . . . . . . 14 ({𝑥, 𝑦} = {𝑤} → 𝑥 = 𝑦)
5249, 51syl 17 . . . . . . . . . . . . 13 ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → 𝑥 = 𝑦)
53 dfsn2 4541 . . . . . . . . . . . . . . . . . . . 20 {𝑥} = {𝑥, 𝑥}
54 preq2 4633 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → {𝑥, 𝑥} = {𝑥, 𝑦})
5553, 54syl5req 2849 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → {𝑥, 𝑦} = {𝑥})
5655eqeq2d 2812 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐴 = {𝑥, 𝑦} ↔ 𝐴 = {𝑥}))
5753, 54syl5eq 2848 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → {𝑥} = {𝑥, 𝑦})
5857eqeq2d 2812 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝐵 = {𝑥} ↔ 𝐵 = {𝑥, 𝑦}))
5956, 58anbi12d 633 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) ↔ (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6059biimpd 232 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6160expd 419 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐴 = {𝑥, 𝑦} → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
6261com12 32 . . . . . . . . . . . . . 14 (𝐴 = {𝑥, 𝑦} → (𝑥 = 𝑦 → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
6362adantr 484 . . . . . . . . . . . . 13 ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → (𝑥 = 𝑦 → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
6452, 63mpd 15 . . . . . . . . . . . 12 ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6564expcom 417 . . . . . . . . . . 11 (𝐴 = {𝑤} → (𝐴 = {𝑥, 𝑦} → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
6665impd 414 . . . . . . . . . 10 (𝐴 = {𝑤} → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6748, 66jaod 856 . . . . . . . . 9 (𝐴 = {𝑤} → (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
6847, 67syl5bi 245 . . . . . . . 8 (𝐴 = {𝑤} → ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
69682eximdv 1920 . . . . . . 7 (𝐴 = {𝑤} → (∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
7069exlimiv 1931 . . . . . 6 (∃𝑤 𝐴 = {𝑤} → (∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
7170imp 410 . . . . 5 ((∃𝑤 𝐴 = {𝑤} ∧ ∃𝑥𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩) → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
7236, 44, 71syl2an 598 . . . 4 ((∀𝑧(𝑧 = {𝐴} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)) → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
7313, 72sylbi 220 . . 3 (⟨𝐴, 𝐵⟩ ⊆ (V × V) → ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
74 simpr 488 . . . . . . . . . . 11 ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → 𝑧 = {𝐴})
75 equid 2019 . . . . . . . . . . . . . 14 𝑥 = 𝑥
7675jctl 527 . . . . . . . . . . . . 13 (𝐴 = {𝑥} → (𝑥 = 𝑥𝐴 = {𝑥}))
7718, 18opeqsn 5362 . . . . . . . . . . . . 13 (⟨𝑥, 𝑥⟩ = {𝐴} ↔ (𝑥 = 𝑥𝐴 = {𝑥}))
7876, 77sylibr 237 . . . . . . . . . . . 12 (𝐴 = {𝑥} → ⟨𝑥, 𝑥⟩ = {𝐴})
7978adantr 484 . . . . . . . . . . 11 ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → ⟨𝑥, 𝑥⟩ = {𝐴})
8074, 79eqtr4d 2839 . . . . . . . . . 10 ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → 𝑧 = ⟨𝑥, 𝑥⟩)
81 opeq12 4770 . . . . . . . . . . . 12 ((𝑤 = 𝑥𝑣 = 𝑥) → ⟨𝑤, 𝑣⟩ = ⟨𝑥, 𝑥⟩)
8281eqeq2d 2812 . . . . . . . . . . 11 ((𝑤 = 𝑥𝑣 = 𝑥) → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑥, 𝑥⟩))
8318, 18, 82spc2ev 3559 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑥⟩ → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
8480, 83syl 17 . . . . . . . . 9 ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
8584adantlr 714 . . . . . . . 8 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴}) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
86 preq12 4634 . . . . . . . . . . . 12 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → {𝐴, 𝐵} = {{𝑥}, {𝑥, 𝑦}})
8786eqeq2d 2812 . . . . . . . . . . 11 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝑧 = {𝐴, 𝐵} ↔ 𝑧 = {{𝑥}, {𝑥, 𝑦}}))
8887biimpa 480 . . . . . . . . . 10 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → 𝑧 = {{𝑥}, {𝑥, 𝑦}})
8918, 19dfop 4765 . . . . . . . . . 10 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
9088, 89eqtr4di 2854 . . . . . . . . 9 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → 𝑧 = ⟨𝑥, 𝑦⟩)
91 opeq12 4770 . . . . . . . . . . 11 ((𝑤 = 𝑥𝑣 = 𝑦) → ⟨𝑤, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
9291eqeq2d 2812 . . . . . . . . . 10 ((𝑤 = 𝑥𝑣 = 𝑦) → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
9318, 19, 92spc2ev 3559 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
9490, 93syl 17 . . . . . . . 8 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
9585, 94jaodan 955 . . . . . . 7 (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ (𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵})) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
9695ex 416 . . . . . 6 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩))
97 elvv 5594 . . . . . 6 (𝑧 ∈ (V × V) ↔ ∃𝑤𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
9896, 5, 973imtr4g 299 . . . . 5 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)))
9998ssrdv 3924 . . . 4 ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ⟨𝐴, 𝐵⟩ ⊆ (V × V))
10099exlimivv 1933 . . 3 (∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ⟨𝐴, 𝐵⟩ ⊆ (V × V))
10173, 100impbii 212 . 2 (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
1021, 101bitri 278 1 (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  wal 1536   = wceq 1538  wex 1781  wcel 2112  Vcvv 3444  wss 3884  {csn 4528  {cpr 4530  cop 4534   × cxp 5521  Rel wrel 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-xp 5529  df-rel 5530
This theorem is referenced by:  funopg  6362
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