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Theorem eliniseg2 5646
Description: Eliminate the class existence constraint in eliniseg 5635. (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 5644 . . 3 Rel 𝐴
2 elrelimasn 5630 . . 3 (Rel 𝐴 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
31, 2ax-mp 5 . 2 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
4 relbrcnvg 5645 . 2 (Rel 𝐴 → (𝐵𝐴𝐶𝐶𝐴𝐵))
53, 4syl5bb 272 1 (Rel 𝐴 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2144  {csn 4314   class class class wbr 4784  ccnv 5248  cima 5252  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262
This theorem is referenced by:  isunit  18864  frege133d  38576
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