Theorem sbthlem6 7644

Description: Lemma for sbth 7649. (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 ) ) }
sbthlem.3	|- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )

Assertion

Ref	Expression
sbthlem6	|- ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )

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

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

Proof of Theorem sbthlem6

Step	Hyp	Ref	Expression
1		df-ima 5018	. . . . 5 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
2		sbthlem.1	. . . . . 6 |- A e. _V
3		sbthlem.2	. . . . . 6 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
4	2, 3	sbthlem4 7642	. . . . 5 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
5	1, 4	syl5reqr 2523	. . . 4 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
6	5	uneq2d 3663	. . 3 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) ) )
7		rnun 5420	. . . 4 |- ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) = ( ran ( f |` U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
8		sbthlem.3	. . . . 5 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
9	8	rneqi 5235	. . . 4 |- ran H = ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
10		df-ima 5018	. . . . 5 |- ( f " U. D ) = ran ( f |` U. D )
11	10	uneq1i 3659	. . . 4 |- ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) ) = ( ran ( f |` U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
12	7, 9, 11	3eqtr4i 2506	. . 3 |- ran H = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
13	6, 12	syl6reqr 2527	. 2 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ran H = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) )
14		imassrn 5354	. . . 4 |- ( f " U. D ) C_ ran f
15		sstr2 3516	. . . 4 |- ( ( f " U. D ) C_ ran f -> ( ran f C_ B -> ( f " U. D ) C_ B ) )
16	14, 15	ax-mp 5	. . 3 |- ( ran f C_ B -> ( f " U. D ) C_ B )
17		undif 3913	. . 3 |- ( ( f " U. D ) C_ B <-> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = B )
18	16, 17	sylib 196	. 2 |- ( ran f C_ B -> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = B )
19	13, 18	sylan9eqr 2530	1 |- ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   {cab 2452   _Vcvv 3118   \ cdif 3478   u. cun 3479   C_ wss 3481   U.cuni 4251   `'ccnv 5004   dom cdm 5005   ran crn 5006   |` cres 5007   "cima 5008   Fun wfun 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-fun 5596

This theorem is referenced by:  sbthlem9  7647