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Theorem elpredg 5837
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
Assertion
Ref Expression
elpredg ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))

Proof of Theorem elpredg
StepHypRef Expression
1 df-pred 5823 . . . . 5 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3952 . . . 4 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
32baib 525 . . 3 (𝑌𝐴 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
43adantl 467 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (𝑅 “ {𝑋})))
5 elimasng 5632 . . 3 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅))
6 df-br 4787 . . 3 (𝑋𝑅𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑅)
75, 6syl6bbr 278 . 2 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑋𝑅𝑌))
8 brcnvg 5441 . 2 ((𝑋𝐵𝑌𝐴) → (𝑋𝑅𝑌𝑌𝑅𝑋))
94, 7, 83bitrd 294 1 ((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wcel 2145  {csn 4316  cop 4322   class class class wbr 4786  ccnv 5248  cima 5252  Predcpred 5822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823
This theorem is referenced by:  predpo  5841  predpoirr  5851  predfrirr  5852  wfrlem10  7577  wsuclem  32107  wsuclb  32110
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