Theorem nfsup 7058

Description: Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)

Hypotheses

Ref	Expression
nfsup.1	|- F/_ x A
nfsup.2	|- F/_ x B
nfsup.3	|- F/_ x R

Assertion

Ref	Expression
nfsup	|- F/_ x sup ( A , B , R )

Proof of Theorem nfsup

Dummy variables u v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sup 7050	. 2 |- sup ( A , B , R ) = U. { u e. B | ( A. v e. A -. u R v /\ A. v e. B ( v R u -> E. w e. A v R w ) ) }
2		nfsup.1	. . . . . 6 |- F/_ x A
3		nfcv 2339	. . . . . . . 8 |- F/_ x u
4		nfsup.3	. . . . . . . 8 |- F/_ x R
5		nfcv 2339	. . . . . . . 8 |- F/_ x v
6	3, 4, 5	nfbr 4079	. . . . . . 7 |- F/ x u R v
7	6	nfn 1672	. . . . . 6 |- F/ x -. u R v
8	2, 7	nfralya 2537	. . . . 5 |- F/ x A. v e. A -. u R v
9		nfsup.2	. . . . . 6 |- F/_ x B
10	5, 4, 3	nfbr 4079	. . . . . . 7 |- F/ x v R u
11		nfcv 2339	. . . . . . . . 9 |- F/_ x w
12	5, 4, 11	nfbr 4079	. . . . . . . 8 |- F/ x v R w
13	2, 12	nfrexya 2538	. . . . . . 7 |- F/ x E. w e. A v R w
14	10, 13	nfim 1586	. . . . . 6 |- F/ x ( v R u -> E. w e. A v R w )
15	9, 14	nfralya 2537	. . . . 5 |- F/ x A. v e. B ( v R u -> E. w e. A v R w )
16	8, 15	nfan 1579	. . . 4 |- F/ x ( A. v e. A -. u R v /\ A. v e. B ( v R u -> E. w e. A v R w ) )
17	16, 9	nfrabw 2678	. . 3 |- F/_ x { u e. B | ( A. v e. A -. u R v /\ A. v e. B ( v R u -> E. w e. A v R w ) ) }
18	17	nfuni 3845	. 2 |- F/_ x U. { u e. B | ( A. v e. A -. u R v /\ A. v e. B ( v R u -> E. w e. A v R w ) ) }
19	1, 18	nfcxfr 2336	1 |- F/_ x sup ( A , B , R )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   F/_wnfc 2326   A.wral 2475   E.wrex 2476   {crab 2479   U.cuni 3839   class class class wbr 4033   supcsup 7048

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-sup 7050

This theorem is referenced by:  nfinf  7083  infssuzcldc  10325