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Theorem relopabi 5835
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5302, ax-nul 5312, ax-pr 5438. (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 . . . . . . 7 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 df-opab 5211 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31, 2eqtri 2763 . . . . . 6 𝐴 = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
43eqabri 2883 . . . . 5 (𝑧𝐴 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 simpl 482 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
652eximi 1833 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
74, 6sylbi 217 . . . 4 (𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
8 ax6evr 2012 . . . . . . . . 9 𝑢 𝑦 = 𝑢
9 pm3.21 471 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ = 𝑧 → (𝑦 = 𝑢 → (𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
109eximdv 1915 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ = 𝑧 → (∃𝑢 𝑦 = 𝑢 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
118, 10mpi 20 . . . . . . . 8 (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧))
12 opeq2 4879 . . . . . . . . . 10 (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
13 eqtr2 2759 . . . . . . . . . . 11 ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ⟨𝑥, 𝑢⟩ = 𝑧)
1413eqcomd 2741 . . . . . . . . . 10 ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
1512, 14sylan 580 . . . . . . . . 9 ((𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
1615eximi 1832 . . . . . . . 8 (∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
1711, 16syl 17 . . . . . . 7 (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
1817eqcoms 2743 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
19182eximi 1833 . . . . 5 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
20 excomim 2161 . . . . 5 (∃𝑥𝑦𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
2119, 20syl 17 . . . 4 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
22 vex 3482 . . . . . . . . 9 𝑥 ∈ V
23 vex 3482 . . . . . . . . 9 𝑢 ∈ V
2422, 23pm3.2i 470 . . . . . . . 8 (𝑥 ∈ V ∧ 𝑢 ∈ V)
2524jctr 524 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑢⟩ → (𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
26252eximi 1833 . . . . . 6 (∃𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
27 df-xp 5695 . . . . . . . 8 (V × V) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)}
28 df-opab 5211 . . . . . . . 8 {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)} = {𝑧 ∣ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
2927, 28eqtri 2763 . . . . . . 7 (V × V) = {𝑧 ∣ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
3029eqabri 2883 . . . . . 6 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
3126, 30sylibr 234 . . . . 5 (∃𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → 𝑧 ∈ (V × V))
3231eximi 1832 . . . 4 (∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦 𝑧 ∈ (V × V))
33 ax5e 1910 . . . 4 (∃𝑦 𝑧 ∈ (V × V) → 𝑧 ∈ (V × V))
347, 21, 32, 334syl 19 . . 3 (𝑧𝐴𝑧 ∈ (V × V))
3534ssriv 3999 . 2 𝐴 ⊆ (V × V)
36 df-rel 5696 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
3735, 36mpbir 231 1 Rel 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  Vcvv 3478  wss 3963  cop 4637  {copab 5210   × cxp 5687  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  relopab  5837  relttrcl  9750  erclwwlkrel  30046  erclwwlknrel  30095
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