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Theorem elcnvintab 36710
Description: Two ways of saying a set is an element of the converse of the intersection of a class. (Contributed by RP, 19-Aug-2020.)
Assertion
Ref Expression
elcnvintab (𝐴 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elcnvintab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . 3 (𝑦 ∈ (V × V) ↦ ⟨(2nd𝑦), (1st𝑦)⟩) = (𝑦 ∈ (V × V) ↦ ⟨(2nd𝑦), (1st𝑦)⟩)
21elcnvlem 36709 . 2 (𝐴 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd𝑦), (1st𝑦)⟩)‘𝐴) ∈ {𝑥𝜑}))
31elcnvlem 36709 . 2 (𝐴𝑥 ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd𝑦), (1st𝑦)⟩)‘𝐴) ∈ 𝑥))
42, 3elmapintab 36704 1 (𝐴 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wcel 1976  {cab 2595  Vcvv 3172  cop 4130   cint 4404  cmpt 4637   × cxp 5025  ccnv 5026  cfv 5789  1st c1st 7034  2nd c2nd 7035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-iota 5753  df-fun 5791  df-fv 5797  df-1st 7036  df-2nd 7037
This theorem is referenced by:  cnvintabd  36711
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