Theorem brub 32061

Description: Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)

Hypotheses

Ref	Expression
brub.1	|- S e. _V
brub.2	|- A e. _V

Assertion

Ref	Expression
brub	|- ( SUB R A <-> A. x e. S x R A )

Distinct variable groups:   x, A   x, R   x, S

Proof of Theorem brub

Step	Hyp	Ref	Expression
1		brub.1	. . . . 5 |- S e. _V
2		brub.2	. . . . 5 |- A e. _V
3		brxp 5147	. . . . 5 |- ( S ( _V X. _V ) A <-> ( S e. _V /\ A e. _V ) )
4	1, 2, 3	mpbir2an 955	. . . 4 |- S ( _V X. _V ) A
5		brdif 4705	. . . 4 |- ( S ( ( _V X. _V ) \ ( ( _V \ R ) o. `' _E ) ) A <-> ( S ( _V X. _V ) A /\ -. S ( ( _V \ R ) o. `' _E ) A ) )
6	4, 5	mpbiran 953	. . 3 |- ( S ( ( _V X. _V ) \ ( ( _V \ R ) o. `' _E ) ) A <-> -. S ( ( _V \ R ) o. `' _E ) A )
7	1, 2	coepr 31642	. . 3 |- ( S ( ( _V \ R ) o. `' _E ) A <-> E. x e. S x ( _V \ R ) A )
8	6, 7	xchbinx 324	. 2 |- ( S ( ( _V X. _V ) \ ( ( _V \ R ) o. `' _E ) ) A <-> -. E. x e. S x ( _V \ R ) A )
9		df-ub 31983	. . 3 |- UB R = ( ( _V X. _V ) \ ( ( _V \ R ) o. `' _E ) )
10	9	breqi 4659	. 2 |- ( SUB R A <-> S ( ( _V X. _V ) \ ( ( _V \ R ) o. `' _E ) ) A )
11		brv 4941	. . . . . 6 |- x _V A
12		brdif 4705	. . . . . 6 |- ( x ( _V \ R ) A <-> ( x _V A /\ -. x R A ) )
13	11, 12	mpbiran 953	. . . . 5 |- ( x ( _V \ R ) A <-> -. x R A )
14	13	rexbii 3041	. . . 4 |- ( E. x e. S x ( _V \ R ) A <-> E. x e. S -. x R A )
15		rexnal 2995	. . . 4 |- ( E. x e. S -. x R A <-> -. A. x e. S x R A )
16	14, 15	bitri 264	. . 3 |- ( E. x e. S x ( _V \ R ) A <-> -. A. x e. S x R A )
17	16	con2bii 347	. 2 |- ( A. x e. S x R A <-> -. E. x e. S x ( _V \ R ) A )
18	8, 10, 17	3bitr4i 292	1 |- ( SUB R A <-> A. x e. S x R A )

Syntax hints:   -. wn 3   <-> wb 196   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   class class class wbr 4653   _E cep 5028   X. cxp 5112   `'ccnv 5113   o. ccom 5118  UBcub 31959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-xp 5120  df-cnv 5122  df-co 5123  df-ub 31983

This theorem is referenced by:  brlb  32062