Theorem eliniseg2 6067

Description: Eliminate the class existence constraint in eliniseg 6055. (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 6065	. . 3 ⊢ Rel ◡𝐴
2		elrelimasn 6047	. . 3 ⊢ (Rel ◡𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶)
4		relbrcnvg 6066	. 2 ⊢ (Rel 𝐴 → (𝐵◡𝐴𝐶 ↔ 𝐶𝐴𝐵))
5	3, 4	bitrid 283	1 ⊢ (Rel 𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  {csn 4568   class class class wbr 5086  ^◡ccnv 5625   “ cima 5629  Rel wrel 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5632  df-rel 5633  df-cnv 5634  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639

This theorem is referenced by:  isunit  20348  frege133d  44214