Theorem sbthlem6 7431

Description: Lemma for sbth 7436. (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 4858	. . . . 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 7429	. . . . 5 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
5	1, 4	syl5reqr 2490	. . . 4 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
6	5	uneq2d 3515	. . 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 5071	. . . 4 |- ran H = ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
10		df-ima 4858	. . . . 5 |- ( f " U. D ) = ran ( f |` U. D )
11	10	uneq1i 3511	. . . 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 2473	. . 3 |- ran H = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
13	6, 12	syl6reqr 2494	. 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 3368	. . . 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 3764	. . 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 2497	1 |- ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   {cab 2429   _Vcvv 2977   \ cdif 3330   u. cun 3331   C_ wss 3333   U.cuni 4096   `'ccnv 4844   dom cdm 4845   ran crn 4846   |` cres 4847   "cima 4848   Fun wfun 5417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5425

This theorem is referenced by:  sbthlem9  7434