Theorem nfsup 7190

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 7182	. 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 2374	. . . . . . . 8 |- F/_ x u
4		nfsup.3	. . . . . . . 8 |- F/_ x R
5		nfcv 2374	. . . . . . . 8 |- F/_ x v
6	3, 4, 5	nfbr 4135	. . . . . . 7 |- F/ x u R v
7	6	nfn 1706	. . . . . 6 |- F/ x -. u R v
8	2, 7	nfralya 2572	. . . . 5 |- F/ x A. v e. A -. u R v
9		nfsup.2	. . . . . 6 |- F/_ x B
10	5, 4, 3	nfbr 4135	. . . . . . 7 |- F/ x v R u
11		nfcv 2374	. . . . . . . . 9 |- F/_ x w
12	5, 4, 11	nfbr 4135	. . . . . . . 8 |- F/ x v R w
13	2, 12	nfrexya 2573	. . . . . . 7 |- F/ x E. w e. A v R w
14	10, 13	nfim 1620	. . . . . 6 |- F/ x ( v R u -> E. w e. A v R w )
15	9, 14	nfralya 2572	. . . . 5 |- F/ x A. v e. B ( v R u -> E. w e. A v R w )
16	8, 15	nfan 1613	. . . 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 2714	. . 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 3899	. 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 2371	1 |- F/_ x sup ( A , B , R )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   F/_wnfc 2361   A.wral 2510   E.wrex 2511   {crab 2514   U.cuni 3893   class class class wbr 4088   supcsup 7180

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-sup 7182

This theorem is referenced by:  nfinf  7215  infssuzcldc  10494