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Theorem cnvssOLD 5200
Description: Obsolete proof of cnvss 5199 as of 27-Apr-2021. Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
cnvssOLD (𝐴𝐵𝐴𝐵)

Proof of Theorem cnvssOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3556 . . . 4 (𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ 𝐵))
2 df-br 4573 . . . 4 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
3 df-br 4573 . . . 4 (𝑦𝐵𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
41, 2, 33imtr4g 283 . . 3 (𝐴𝐵 → (𝑦𝐴𝑥𝑦𝐵𝑥))
54ssopab2dv 4914 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
6 df-cnv 5031 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
7 df-cnv 5031 . 2 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
85, 6, 73sstr4g 3603 1 (𝐴𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1975  wss 3534  cop 4125   class class class wbr 4572  {copab 4631  ccnv 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-in 3541  df-ss 3548  df-br 4573  df-opab 4633  df-cnv 5031
This theorem is referenced by: (None)
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