Theorem eliniseg2 6003

Description: Eliminate the class existence constraint in eliniseg 5991. (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

Step	Hyp	Ref	Expression
1		relcnv 6001	. . 3 ⊢ Rel ◡𝐴
2		elrelimasn 5982	. . 3 ⊢ (Rel ◡𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶)
4		relbrcnvg 6002	. 2 ⊢ (Rel 𝐴 → (𝐵◡𝐴𝐶 ↔ 𝐶𝐴𝐵))
5	3, 4	syl5bb 282	1 ⊢ (Rel 𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2108  {csn 4558   class class class wbr 5070  ^◡ccnv 5579   “ cima 5583  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593

This theorem is referenced by:  isunit  19814  frege133d  41262