Theorem sbthlem4 4450

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))}

Assertion

Ref	Expression
sbthlem4	|- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))

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

Proof of Theorem sbthlem4

Step	Hyp	Ref	Expression
1		difss 2167	. . . . . . 7 |- (B \ (f"U.D)) (_ B
2		sseq2 2083	. . . . . . 7 |- (dom g = B -> ((B \ (f"U.D)) (_ dom g <-> (B \ (f"U.D)) (_ B))
3	1, 2	mpbiri 194	. . . . . 6 |- (dom g = B -> (B \ (f"U.D)) (_ dom g)
4		ssdmres 3381	. . . . . 6 |- ((B \ (f"U.D)) (_ dom g <-> dom ( g |` (B \ (f"U.D))) = (B \ (f"U.D)))
5	3, 4	sylib 198	. . . . 5 |- (dom g = B -> dom ( g |` (B \ (f"U.D))) = (B \ (f"U.D)))
6		dfdm4 3305	. . . . 5 |- dom ( g |` (B \ (f"U.D))) = ran `'(g |` (B \ (f"U.D)))
7	5, 6	syl5reqr 1522	. . . 4 |- (dom g = B -> (B \ (f"U.D)) = ran `'(g |` (B \ (f"U.D))))
8		funcnvres 3568	. . . . . 6 |- (Fun `'g -> `'(g |` (B \ (f"U.D))) = (`'g |` (g"(B \ (f"U.D)))))
9		sbthlem.1	. . . . . . . 8 |- A e. V
10		sbthlem.2	. . . . . . . 8 |- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
11	9, 10	sbthlem3 4449	. . . . . . 7 |- (ran g (_ A -> (g"(B \ (f"U.D))) = (A \ U.D))
12		reseq2 3369	. . . . . . 7 |- ((g"(B \ (f"U.D))) = (A \ U.D) -> (`'g |` (g"(B \ (f"U.D)))) = (`'g |` (A \ U.D)))
13	11, 12	syl 10	. . . . . 6 |- (ran g (_ A -> (`'g |` (g"(B \ (f"U.D)))) = (`'g |` (A \ U.D)))
14	8, 13	sylan9eqr 1529	. . . . 5 |- ((ran g (_ A /\ Fun `'g) -> `'(g |` (B \ (f"U.D))) = (`'g |` (A \ U.D)))
15	14	rneqd 3341	. . . 4 |- ((ran g (_ A /\ Fun `'g) -> ran `'(g |` (B \ (f"U.D))) = ran (`'g |` (A \ U.D)))
16	7, 15	sylan9eq 1527	. . 3 |- ((dom g = B /\ (ran g (_ A /\ Fun `'g)) -> (B \ (f"U.D)) = ran (`'g |` (A \ U.D)))
17	16	anassrs 441	. 2 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (B \ (f"U.D)) = ran (`'g |` (A \ U.D)))
18		df-ima 3191	. 2 |- (`'g"(A \ U.D)) = ran (`'g |` (A \ U.D))
19	17, 18	syl6reqr 1526	1 |- (((dom g = B /\ ran g (_ A) /\ Fun `'g) -> (`'g"(A \ U.D)) = (B \ (f"U.D)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   \ cdif 2044   (_ 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:  sbthlem6 4452  sbthlem8 4454

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

Copyright terms: Public domain