Theorem sbthlem5 4451

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
sbthlem5	|- ((dom f = A /\ ran g (_ A) -> dom H = A)

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

Proof of Theorem sbthlem5

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . . . 9 |- A e. V
2		sbthlem.2	. . . . . . . . 9 |- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
3	1, 2	sbthlem1 4447	. . . . . . . 8 |- U.D (_ (A \ (g"(B \ (f"U.D))))
4		difss 2167	. . . . . . . 8 |- (A \ (g"(B \ (f"U.D)))) (_ A
5	3, 4	sstri 2073	. . . . . . 7 |- U.D (_ A
6		sseq2 2083	. . . . . . 7 |- (dom f = A -> (U.D (_ dom f <-> U.D (_ A))
7	5, 6	mpbiri 194	. . . . . 6 |- (dom f = A -> U.D (_ dom f)
8		dfss 2054	. . . . . 6 |- (U.D (_ dom f <-> U.D = (U.D i^i dom f))
9	7, 8	sylib 198	. . . . 5 |- (dom f = A -> U.D = (U.D i^i dom f))
10	9	uneq1d 2183	. . . 4 |- (dom f = A -> (U.D u. (A \ U.D)) = ((U.D i^i dom f) u. (A \ U.D)))
11		imassrn 3415	. . . . . . 7 |- (g"(B \ (f"U.D))) (_ ran g
12	1, 2	sbthlem3 4449	. . . . . . . 8 |- (ran g (_ A -> (g"(B \ (f"U.D))) = (A \ U.D))
13	12	sseq1d 2088	. . . . . . 7 |- (ran g (_ A -> ((g"(B \ (f"U.D))) (_ ran g <-> (A \ U.D) (_ ran g))
14	11, 13	mpbii 193	. . . . . 6 |- (ran g (_ A -> (A \ U.D) (_ ran g)
15		dfss 2054	. . . . . 6 |- ((A \ U.D) (_ ran g <-> (A \ U.D) = ((A \ U.D) i^i ran g))
16	14, 15	sylib 198	. . . . 5 |- (ran g (_ A -> (A \ U.D) = ((A \ U.D) i^i ran g))
17	16	uneq2d 2184	. . . 4 |- (ran g (_ A -> ((U.D i^i dom f) u. (A \ U.D)) = ((U.D i^i dom f) u. ((A \ U.D) i^i ran g)))
18	10, 17	sylan9eq 1527	. . 3 |- ((dom f = A /\ ran g (_ A) -> (U.D u. (A \ U.D)) = ((U.D i^i dom f) u. ((A \ U.D) i^i ran g)))
19		sbthlem.3	. . . . 5 |- H = ((f |` U.D) u. (`'g |` (A \ U.D)))
20	19	dmeqi 3312	. . . 4 |- dom H = dom ((f |` U.D) u. (`'g |` (A \ U.D)))
21		dmun 3317	. . . 4 |- dom ((f |` U.D) u. (`'g |` (A \ U.D))) = (dom ( f |` U.D) u. dom (`'g |` (A \ U.D)))
22		dmres 3380	. . . . 5 |- dom ( f |` U.D) = (U.D i^i dom f)
23		dmres 3380	. . . . . 6 |- dom (`'g |` (A \ U.D)) = ((A \ U.D) i^i dom `' g)
24		df-rn 3189	. . . . . . . 8 |- ran g = dom `' g
25	24	eqcomi 1479	. . . . . . 7 |- dom `' g = ran g
26	25	ineq2i 2214	. . . . . 6 |- ((A \ U.D) i^i dom `' g) = ((A \ U.D) i^i ran g)
27	23, 26	eqtr 1495	. . . . 5 |- dom (`'g |` (A \ U.D)) = ((A \ U.D) i^i ran g)
28	22, 27	uneq12i 2182	. . . 4 |- (dom ( f |` U.D) u. dom (`'g |` (A \ U.D))) = ((U.D i^i dom f) u. ((A \ U.D) i^i ran g))
29	20, 21, 28	3eqtr 1499	. . 3 |- dom H = ((U.D i^i dom f) u. ((A \ U.D) i^i ran g))
30	18, 29	syl6reqr 1526	. 2 |- ((dom f = A /\ ran g (_ A) -> dom H = (U.D u. (A \ U.D)))
31		undif 2343	. . 3 |- (U.D (_ A <-> (U.D u. (A \ U.D)) = A)
32	5, 31	mpbi 189	. 2 |- (U.D u. (A \ U.D)) = A
33	30, 32	syl6eq 1523	1 |- ((dom f = A /\ ran g (_ A) -> dom H = A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   \ cdif 2044   u. cun 2045   i^i cin 2046   (_ wss 2047  U.cuni 2503  `'ccnv 3169  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173

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-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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