Theorem eliniseg2 5063

Description: Eliminate the class existence constraint in eliniseg 5053. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)

Assertion

Ref	Expression
eliniseg2	|- ( Rel A -> ( C e. ( `' A " { B } ) <-> C A B ) )

Proof of Theorem eliniseg2

Step	Hyp	Ref	Expression
1		relcnv 5061	. . 3 |- Rel `' A
2		elrelimasn 5049	. . 3 |- ( Rel `' A -> ( C e. ( `' A " { B } ) <-> B `' A C ) )
3	1, 2	ax-mp 5	. 2 |- ( C e. ( `' A " { B } ) <-> B `' A C )
4		relbrcnvg 5062	. 2 |- ( Rel A -> ( B `' A C <-> C A B ) )
5	3, 4	bitrid 192	1 |- ( Rel A -> ( C e. ( `' A " { B } ) <-> C A B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2176   {csn 3633   class class class wbr 4045   `'ccnv 4675   "cima 4679   Rel wrel 4681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683  df-cnv 4684  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689

This theorem is referenced by:  isunitd  13901