Theorem sbthlem3 8072

Description: Lemma for sbth 8080. (Contributed by NM, 22-Mar-1998.)

Hypotheses

Ref	Expression
sbthlem.1	|- A e. _V
sbthlem.2	|- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }

Assertion

Ref	Expression
sbthlem3	|- ( ran g C_ A -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )

Distinct variable groups:   x, A   x, B   x, D   x, f   x, g

Allowed substitution hints:   A( f, g)   B( f, g)   D( f, g)

Proof of Theorem sbthlem3

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . 6 |- A e. _V
2		sbthlem.2	. . . . . 6 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
3	1, 2	sbthlem2 8071	. . . . 5 |- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
4	1, 2	sbthlem1 8070	. . . . 5 |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )
5	3, 4	jctil 560	. . . 4 |- ( ran g C_ A -> ( U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) /\ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D ) )
6		eqss 3618	. . . 4 |- ( U. D = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) <-> ( U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) /\ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D ) )
7	5, 6	sylibr 224	. . 3 |- ( ran g C_ A -> U. D = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
8	7	difeq2d 3728	. 2 |- ( ran g C_ A -> ( A \ U. D ) = ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
9		imassrn 5477	. . . 4 |- ( g " ( B \ ( f " U. D ) ) ) C_ ran g
10		sstr2 3610	. . . 4 |- ( ( g " ( B \ ( f " U. D ) ) ) C_ ran g -> ( ran g C_ A -> ( g " ( B \ ( f " U. D ) ) ) C_ A ) )
11	9, 10	ax-mp 5	. . 3 |- ( ran g C_ A -> ( g " ( B \ ( f " U. D ) ) ) C_ A )
12		dfss4 3858	. . 3 |- ( ( g " ( B \ ( f " U. D ) ) ) C_ A <-> ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) = ( g " ( B \ ( f " U. D ) ) ) )
13	11, 12	sylib 208	. 2 |- ( ran g C_ A -> ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) = ( g " ( B \ ( f " U. D ) ) ) )
14	8, 13	eqtr2d 2657	1 |- ( ran g C_ A -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   \ cdif 3571   C_ wss 3574   U.cuni 4436   ran crn 5115   "cima 5117

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-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-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  sbthlem4  8073  sbthlem5  8074