Theorem sbthlem4 8073

Description: Lemma for sbth 8080. (Contributed by NM, 27-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
sbthlem4	|- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " 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 sbthlem4

Step	Hyp	Ref	Expression
1		dfdm4 5316	. . . . 5 |- dom ( g |` ( B \ ( f " U. D ) ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) )
2		difss 3737	. . . . . . 7 |- ( B \ ( f " U. D ) ) C_ B
3		sseq2 3627	. . . . . . 7 |- ( dom g = B -> ( ( B \ ( f " U. D ) ) C_ dom g <-> ( B \ ( f " U. D ) ) C_ B ) )
4	2, 3	mpbiri 248	. . . . . 6 |- ( dom g = B -> ( B \ ( f " U. D ) ) C_ dom g )
5		ssdmres 5420	. . . . . 6 |- ( ( B \ ( f " U. D ) ) C_ dom g <-> dom ( g |` ( B \ ( f " U. D ) ) ) = ( B \ ( f " U. D ) ) )
6	4, 5	sylib 208	. . . . 5 |- ( dom g = B -> dom ( g |` ( B \ ( f " U. D ) ) ) = ( B \ ( f " U. D ) ) )
7	1, 6	syl5reqr 2671	. . . 4 |- ( dom g = B -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
8		funcnvres 5967	. . . . . 6 |- ( Fun `' g -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) )
9		sbthlem.1	. . . . . . . 8 |- A e. _V
10		sbthlem.2	. . . . . . . 8 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
11	9, 10	sbthlem3 8072	. . . . . . 7 |- ( ran g C_ A -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )
12	11	reseq2d 5396	. . . . . 6 |- ( ran g C_ A -> ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) = ( `' g |` ( A \ U. D ) ) )
13	8, 12	sylan9eqr 2678	. . . . 5 |- ( ( ran g C_ A /\ Fun `' g ) -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( A \ U. D ) ) )
14	13	rneqd 5353	. . . 4 |- ( ( ran g C_ A /\ Fun `' g ) -> ran `' ( g |` ( B \ ( f " U. D ) ) ) = ran ( `' g |` ( A \ U. D ) ) )
15	7, 14	sylan9eq 2676	. . 3 |- ( ( dom g = B /\ ( ran g C_ A /\ Fun `' g ) ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
16	15	anassrs 680	. 2 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
17		df-ima 5127	. 2 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
18	16, 17	syl6reqr 2675	1 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " 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   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882

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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890

This theorem is referenced by:  sbthlem6  8075  sbthlem8  8077