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Theorem relopabi 5447
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 4975, ax-nul 4983, ax-pr 5096. (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 4907 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31, 2eqtri 2828 . . . . . . 7 𝐴 = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
43abeq2i 2919 . . . . . 6 (𝑧𝐴 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 simpl 470 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
652eximi 1920 . . . . . 6 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
74, 6sylbi 208 . . . . 5 (𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
8 ax6evr 2111 . . . . . . . . . 10 𝑢 𝑦 = 𝑢
9 pm3.21 459 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = 𝑧 → (𝑦 = 𝑢 → (𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
109eximdv 2008 . . . . . . . . . 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 4596 . . . . . . . . . . 11 (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
13 eqtr2 2826 . . . . . . . . . . . 12 ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ⟨𝑥, 𝑢⟩ = 𝑧)
1413eqcomd 2812 . . . . . . . . . . 11 ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
1512, 14sylan 571 . . . . . . . . . 10 ((𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
1615eximi 1919 . . . . . . . . 9 (∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
1711, 16syl 17 . . . . . . . 8 (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
1817eqcoms 2814 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
19182eximi 1920 . . . . . 6 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
20 excomim 2210 . . . . . 6 (∃𝑥𝑦𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
2119, 20syl 17 . . . . 5 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
22 vex 3394 . . . . . . . . . 10 𝑥 ∈ V
23 vex 3394 . . . . . . . . . 10 𝑢 ∈ V
2422, 23pm3.2i 458 . . . . . . . . 9 (𝑥 ∈ V ∧ 𝑢 ∈ V)
2524jctr 516 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑢⟩ → (𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
26252eximi 1920 . . . . . . 7 (∃𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
27 df-xp 5317 . . . . . . . . 9 (V × V) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)}
28 df-opab 4907 . . . . . . . . 9 {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)} = {𝑧 ∣ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
2927, 28eqtri 2828 . . . . . . . 8 (V × V) = {𝑧 ∣ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
3029abeq2i 2919 . . . . . . 7 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
3126, 30sylibr 225 . . . . . 6 (∃𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → 𝑧 ∈ (V × V))
3231eximi 1919 . . . . 5 (∃𝑦𝑥𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦 𝑧 ∈ (V × V))
337, 21, 323syl 18 . . . 4 (𝑧𝐴 → ∃𝑦 𝑧 ∈ (V × V))
34 ax5e 2003 . . . 4 (∃𝑦 𝑧 ∈ (V × V) → 𝑧 ∈ (V × V))
3533, 34syl 17 . . 3 (𝑧𝐴𝑧 ∈ (V × V))
3635ssriv 3802 . 2 𝐴 ⊆ (V × V)
37 df-rel 5318 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
3836, 37mpbir 222 1 Rel 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wex 1859  wcel 2156  {cab 2792  Vcvv 3391  wss 3769  cop 4376  {copab 4906   × cxp 5309  Rel wrel 5316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-opab 4907  df-xp 5317  df-rel 5318
This theorem is referenced by:  relopab  5449  mptrel  5450  reli  5451  rele  5452  relcnv  5713  cotrg  5717  relco  5847  brfvopabrbr  6496  reloprab  6928  reldmoprab  6971  relrpss  7164  eqer  8010  ecopover  8083  relen  8193  reldom  8194  relfsupp  8512  relwdom  8706  fpwwe2lem2  9735  fpwwe2lem3  9736  fpwwe2lem6  9738  fpwwe2lem7  9739  fpwwe2lem9  9741  fpwwe2lem11  9743  fpwwe2lem12  9744  fpwwe2lem13  9745  fpwwelem  9748  climrel  14442  rlimrel  14443  brstruct  16073  sscrel  16673  gaorber  17938  sylow2a  18231  efgrelexlemb  18360  efgcpbllemb  18365  rellindf  20353  2ndcctbss  21468  refrel  21521  vitalilem1  23585  lgsquadlem1  25315  lgsquadlem2  25316  relsubgr  26373  erclwwlkrel  27156  erclwwlknrel  27213  vcrel  27739  h2hlm  28161  hlimi  28369  relmntop  30389  relae  30624  dmscut  32234  fnerel  32649  filnetlem3  32691  cnfinltrel  33552  brabg2  33817  heiborlem3  33918  heiborlem4  33919  relrngo  34001  isdivrngo  34055  drngoi  34056  isdrngo1  34061  riscer  34093  relcoss  34486  relssr  34558  prter1  34653  prter3  34656  reldvds  39008  nelbrim  41858  rellininds  42794
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