Theorem eliniseg2 6108

Description: Eliminate the class existence constraint in eliniseg 6098. (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 6106	. . 3 ⊢ Rel ◡𝐴
2		elrelimasn 6090	. . 3 ⊢ (Rel ◡𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶)
4		relbrcnvg 6107	. 2 ⊢ (Rel 𝐴 → (𝐵◡𝐴𝐶 ↔ 𝐶𝐴𝐵))
5	3, 4	bitrid 286	1 ⊢ (Rel 𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2146  {csn 4588   class class class wbr 5106  ^◡ccnv 5654   “ cima 5658  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  isunit  20618  frege133d  44908