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Theorem eliniseg2 5410
Description: Eliminate the class existence constraint in eliniseg 5399. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)
Assertion
Ref Expression
eliniseg2 (Rel 𝐴 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Proof of Theorem eliniseg2
StepHypRef Expression
1 relcnv 5408 . . 3 Rel 𝐴
2 elrelimasn 5394 . . 3 (Rel 𝐴 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
31, 2ax-mp 5 . 2 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
4 relbrcnvg 5409 . 2 (Rel 𝐴 → (𝐵𝐴𝐶𝐶𝐴𝐵))
53, 4syl5bb 270 1 (Rel 𝐴 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wcel 1976  {csn 4124   class class class wbr 4577  ccnv 5026  cima 5030  Rel wrel 5032
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-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
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-ne 2781  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-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040
This theorem is referenced by:  isunit  18428  frege133d  36859
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