Theorem sbthlem6 4452

Description: Lemma for sbth 4457.

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 4450	. . . . 5 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))
4		df-ima 3191	. . . . 5 |- (`'g"(A \ U.D)) = ran (`'g |` (A \ U.D))
5	3, 4	syl5reqr 1522	. . . 4 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (B \ (f"U.D)) = ran (`'g |` (A \ U.D)))
6	5	uneq2d 2184	. . 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 3457	. . . 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 3340	. . . 4 |- ran H = ran ((f |` U.D) u. (`'g |` (A \ U.D)))
10		df-ima 3191	. . . . 5 |- (f"U.D) = ran ( f |` U.D)
11	10	uneq1i 2180	. . . 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 1505	. . 3 |- ran H = ((f"U.D) u. ran (`'g |` (A \ U.D)))
13	6, 12	syl6reqr 1526	. 2 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> ran H = ((f"U.D) u. (B \ (f"U.D))))
14		imassrn 3415	. . . 4 |- (f"U.D) (_ ran f
15		sstr2 2071	. . . 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 2343	. . 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 1529	1 |- ((ran f (_ B /\ ((dom g = B /\ ran g (_ A) /\ Fun `'g)) -> ran H = B)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   \ cdif 2044   u. cun 2045   (_ wss 2047  U.cuni 2503  `'ccnv 3169  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173  Fun wfun 3176

This theorem is referenced by:  sbthlem9 4455

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192

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