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Theorem relopabi 5818
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5293, ax-nul 5300, ax-pr 5423. (Revised by KP, 25-Oct-2021.)
Hypothesis
Ref Expression
relopabi.1 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
relopabi Rel 𝐴

Proof of Theorem relopabi
Dummy variables 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopabi.1 . . . . . . . 8 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 df-opab 5205 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31, 2eqtri 2755 . . . . . . 7 𝐴 = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
43eqabri 2872 . . . . . 6 (𝑧𝐴 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 simpl 482 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
652eximi 1831 . . . . . 6 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
74, 6sylbi 216 . . . . 5 (𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
8 ax6evr 2011 . . . . . . . . . 10 𝑢 𝑦 = 𝑢
9 pm3.21 471 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = 𝑧 → (𝑦 = 𝑢 → (𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
109eximdv 1913 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ = 𝑧 → (∃𝑢 𝑦 = 𝑢 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
118, 10mpi 20 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧))
12 opeq2 4870 . . . . . . . . . . 11 (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
13 eqtr2 2751 . . . . . . . . . . . 12 ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ⟨𝑥, 𝑢⟩ = 𝑧)
1413eqcomd 2733 . . . . . . . . . . 11 ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
1512, 14sylan 579 . . . . . . . . . 10 ((𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
1615eximi 1830 . . . . . . . . 9 (∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
1711, 16syl 17 . . . . . . . 8 (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
1817eqcoms 2735 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
19182eximi 1831 . . . . . 6 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
20 excomim 2156 . . . . . 6 (∃𝑥𝑦𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
2119, 20syl 17 . . . . 5 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
22 vex 3473 . . . . . . . . . 10 𝑥 ∈ V
23 vex 3473 . . . . . . . . . 10 𝑢 ∈ V
2422, 23pm3.2i 470 . . . . . . . . 9 (𝑥 ∈ V ∧ 𝑢 ∈ V)
2524jctr 524 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑢⟩ → (𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
26252eximi 1831 . . . . . . 7 (∃𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
27 df-xp 5678 . . . . . . . . 9 (V × V) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)}
28 df-opab 5205 . . . . . . . . 9 {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)} = {𝑧 ∣ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
2927, 28eqtri 2755 . . . . . . . 8 (V × V) = {𝑧 ∣ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
3029eqabri 2872 . . . . . . 7 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
3126, 30sylibr 233 . . . . . 6 (∃𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → 𝑧 ∈ (V × V))
3231eximi 1830 . . . . 5 (∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦 𝑧 ∈ (V × V))
337, 21, 323syl 18 . . . 4 (𝑧𝐴 → ∃𝑦 𝑧 ∈ (V × V))
34 ax5e 1908 . . . 4 (∃𝑦 𝑧 ∈ (V × V) → 𝑧 ∈ (V × V))
3533, 34syl 17 . . 3 (𝑧𝐴𝑧 ∈ (V × V))
3635ssriv 3982 . 2 𝐴 ⊆ (V × V)
37 df-rel 5679 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
3836, 37mpbir 230 1 Rel 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wex 1774  wcel 2099  {cab 2704  Vcvv 3469  wss 3944  cop 4630  {copab 5204   × cxp 5670  Rel wrel 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-12 2164  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5205  df-xp 5678  df-rel 5679
This theorem is referenced by:  relopab  5820  relttrcl  9729  erclwwlkrel  29820  erclwwlknrel  29869
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