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Theorem relopabi 5401
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 4933, ax-nul 4941, ax-pr 5055. (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 4865 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31, 2eqtri 2782 . . . . . . 7 𝐴 = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
43abeq2i 2873 . . . . . 6 (𝑧𝐴 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 simpl 474 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
652eximi 1912 . . . . . 6 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
74, 6sylbi 207 . . . . 5 (𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
8 ax6evr 2097 . . . . . . . . . 10 𝑢 𝑦 = 𝑢
9 pm3.21 463 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = 𝑧 → (𝑦 = 𝑢 → (𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
109eximdv 1995 . . . . . . . . . 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 4554 . . . . . . . . . . 11 (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
13 eqtr2 2780 . . . . . . . . . . . 12 ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ⟨𝑥, 𝑢⟩ = 𝑧)
1413eqcomd 2766 . . . . . . . . . . 11 ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
1512, 14sylan 489 . . . . . . . . . 10 ((𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
1615eximi 1911 . . . . . . . . 9 (∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
1711, 16syl 17 . . . . . . . 8 (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
1817eqcoms 2768 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
19182eximi 1912 . . . . . 6 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
20 excomim 2192 . . . . . 6 (∃𝑥𝑦𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
2119, 20syl 17 . . . . 5 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
22 vex 3343 . . . . . . . . . 10 𝑥 ∈ V
23 vex 3343 . . . . . . . . . 10 𝑢 ∈ V
2422, 23pm3.2i 470 . . . . . . . . 9 (𝑥 ∈ V ∧ 𝑢 ∈ V)
2524jctr 566 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑢⟩ → (𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
26252eximi 1912 . . . . . . 7 (∃𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
27 df-xp 5272 . . . . . . . . 9 (V × V) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)}
28 df-opab 4865 . . . . . . . . 9 {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)} = {𝑧 ∣ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
2927, 28eqtri 2782 . . . . . . . 8 (V × V) = {𝑧 ∣ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
3029abeq2i 2873 . . . . . . 7 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
3126, 30sylibr 224 . . . . . 6 (∃𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → 𝑧 ∈ (V × V))
3231eximi 1911 . . . . 5 (∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦 𝑧 ∈ (V × V))
337, 21, 323syl 18 . . . 4 (𝑧𝐴 → ∃𝑦 𝑧 ∈ (V × V))
34 ax5e 1990 . . . 4 (∃𝑦 𝑧 ∈ (V × V) → 𝑧 ∈ (V × V))
3533, 34syl 17 . . 3 (𝑧𝐴𝑧 ∈ (V × V))
3635ssriv 3748 . 2 𝐴 ⊆ (V × V)
37 df-rel 5273 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
3836, 37mpbir 221 1 Rel 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  Vcvv 3340  wss 3715  cop 4327  {copab 4864   × cxp 5264  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by:  relopab  5403  mptrel  5404  reli  5405  rele  5406  relcnv  5661  cotrg  5665  relco  5794  brfvopabrbr  6441  reloprab  6867  reldmoprab  6910  relrpss  7103  eqer  7946  ecopover  8018  relen  8126  reldom  8127  relfsupp  8442  relwdom  8636  fpwwe2lem2  9646  fpwwe2lem3  9647  fpwwe2lem6  9649  fpwwe2lem7  9650  fpwwe2lem9  9652  fpwwe2lem11  9654  fpwwe2lem12  9655  fpwwe2lem13  9656  fpwwelem  9659  climrel  14422  rlimrel  14423  brstruct  16068  sscrel  16674  gaorber  17941  sylow2a  18234  efgrelexlemb  18363  efgcpbllemb  18368  rellindf  20349  2ndcctbss  21460  refrel  21513  vitalilem1  23576  lgsquadlem1  25304  lgsquadlem2  25305  relsubgr  26360  erclwwlkrel  27140  erclwwlknrel  27197  vcrel  27724  h2hlm  28146  hlimi  28354  relmntop  30377  relae  30612  dmscut  32224  fnerel  32639  filnetlem3  32681  cnfinltrel  33552  brabg2  33823  heiborlem3  33925  heiborlem4  33926  relrngo  34008  isdivrngo  34062  drngoi  34063  isdrngo1  34068  riscer  34100  relcoss  34501  relssr  34573  prter1  34668  prter3  34671  reldvds  39016  nelbrim  41801  rellininds  42742
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