Theorem nfsup 7051

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 7043	. 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 2336	. . . . . . . 8 |- F/_ x u
4		nfsup.3	. . . . . . . 8 |- F/_ x R
5		nfcv 2336	. . . . . . . 8 |- F/_ x v
6	3, 4, 5	nfbr 4075	. . . . . . 7 |- F/ x u R v
7	6	nfn 1669	. . . . . 6 |- F/ x -. u R v
8	2, 7	nfralya 2534	. . . . 5 |- F/ x A. v e. A -. u R v
9		nfsup.2	. . . . . 6 |- F/_ x B
10	5, 4, 3	nfbr 4075	. . . . . . 7 |- F/ x v R u
11		nfcv 2336	. . . . . . . . 9 |- F/_ x w
12	5, 4, 11	nfbr 4075	. . . . . . . 8 |- F/ x v R w
13	2, 12	nfrexya 2535	. . . . . . 7 |- F/ x E. w e. A v R w
14	10, 13	nfim 1583	. . . . . 6 |- F/ x ( v R u -> E. w e. A v R w )
15	9, 14	nfralya 2534	. . . . 5 |- F/ x A. v e. B ( v R u -> E. w e. A v R w )
16	8, 15	nfan 1576	. . . 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 2675	. . 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 3841	. 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 2333	1 |- F/_ x sup ( A , B , R )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   F/_wnfc 2323   A.wral 2472   E.wrex 2473   {crab 2476   U.cuni 3835   class class class wbr 4029   supcsup 7041

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-sup 7043

This theorem is referenced by:  nfinf  7076  infssuzcldc  12088