Theorem sbthlem6 7651

Description: Lemma for sbth 7656. (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 5021	. . . . 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 7649	. . . . 5 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
5	1, 4	syl5reqr 2513	. . . 4 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
6	5	uneq2d 3654	. . 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 5421	. . . 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 5239	. . . 4 |- ran H = ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
10		df-ima 5021	. . . . 5 |- ( f " U. D ) = ran ( f |` U. D )
11	10	uneq1i 3650	. . . 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 2496	. . 3 |- ran H = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
13	6, 12	syl6reqr 2517	. 2 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ran H = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) )
14		imassrn 5358	. . . 4 |- ( f " U. D ) C_ ran f
15		sstr2 3506	. . . 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 3911	. . 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 2520	1 |- ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   {cab 2442   _Vcvv 3109   \ cdif 3468   u. cun 3469   C_ wss 3471   U.cuni 4251   `'ccnv 5007   dom cdm 5008   ran crn 5009   |` cres 5010   "cima 5011   Fun wfun 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-fun 5596

This theorem is referenced by:  sbthlem9  7654