Theorem sbthlem6 4458

Description: Lemma for sbth 4463.

Hypotheses

Ref	Expression
sbthlem.1	|- A e. V
sbthlem.2	|- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
sbthlem.3	|- H = ((f |` U.D) u. (`'g |` (A \ U.D)))

Assertion

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

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

Proof of Theorem sbthlem6

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . 6 |- A e. V
2		sbthlem.2	. . . . . 6 |- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
3	1, 2	sbthlem4 4456	. . . . 5 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))
4		df-ima 3197	. . . . 5 |- (`'g"(A \ U.D)) = ran (`'g |` (A \ U.D))
5	3, 4	syl5reqr 1525	. . . 4 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (B \ (f"U.D)) = ran (`'g |` (A \ U.D)))
6	5	uneq2d 2187	. . 3 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> ((f"U.D) u. (B \ (f"U.D))) = ((f"U.D) u. ran (`'g |` (A \ U.D))))
7		rnun 3463	. . . 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 3346	. . . 4 |- ran H = ran ((f |` U.D) u. (`'g |` (A \ U.D)))
10		df-ima 3197	. . . . 5 |- (f"U.D) = ran ( f |` U.D)
11	10	uneq1i 2183	. . . 4 |- ((f"U.D) u. ran (`'g |` (A \ U.D))) = (ran ( f |` U.D) u. ran (`'g |` (A \ U.D)))
12	7, 9, 11	3eqtr4 1508	. . 3 |- ran H = ((f"U.D) u. ran (`'g |` (A \ U.D)))
13	6, 12	syl6reqr 1529	. 2 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> ran H = ((f"U.D) u. (B \ (f"U.D))))
14		imassrn 3421	. . . 4 |- (f"U.D) (_ ran f
15		sstr2 2074	. . . 4 |- ((f"U.D) (_ ran f -> (ran f (_ B -> (f"U.D) (_ B))
16	14, 15	ax-mp 7	. . 3 |- (ran f (_ B -> (f"U.D) (_ B)
17		undif 2347	. . 3 |- ((f"U.D) (_ B <-> ((f"U.D) u. (B \ (f"U.D))) = B)
18	16, 17	sylib 198	. 2 |- (ran f (_ B -> ((f"U.D) u. (B \ (f"U.D))) = B)
19	13, 18	sylan9eqr 1532	1 |- ((ran f (_ B /\ ((dom g = B /\ ran g (_ A) /\ Fun `'g)) -> ran H = B)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  Vcvv 1814   \ cdif 2047   u. cun 2048   (_ wss 2050  U.cuni 2507  `'ccnv 3175  dom cdm 3176  ran crn 3177   |` cres 3178  "cima 3179  Fun wfun 3182

This theorem is referenced by:  sbthlem9 4461

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198

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