Theorem sbthlem6 7705

Description: Lemma for sbth 7710. (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 4852	. . . . 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 7703	. . . . 5 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
5	1, 4	syl5reqr 2520	. . . 4 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
6	5	uneq2d 3579	. . 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 5250	. . . 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 5067	. . . 4 |- ran H = ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
10		df-ima 4852	. . . . 5 |- ( f " U. D ) = ran ( f |` U. D )
11	10	uneq1i 3575	. . . 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 2503	. . 3 |- ran H = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
13	6, 12	syl6reqr 2524	. 2 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ran H = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) )
14		imassrn 5185	. . . 4 |- ( f " U. D ) C_ ran f
15		sstr2 3425	. . . 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 3839	. . 3 |- ( ( f " U. D ) C_ B <-> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = B )
18	16, 17	sylib 201	. 2 |- ( ran f C_ B -> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = B )
19	13, 18	sylan9eqr 2527	1 |- ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   {cab 2457   _Vcvv 3031   \ cdif 3387   u. cun 3388   C_ wss 3390   U.cuni 4190   `'ccnv 4838   dom cdm 4839   ran crn 4840   |` cres 4841   "cima 4842   Fun wfun 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fun 5591

This theorem is referenced by:  sbthlem9  7708