Theorem sbthlem3 4449

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
sbthlem3	|- (ran g (_ A -> (g"(B \ (f"U.D))) = (A \ U.D))

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

Proof of Theorem sbthlem3

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	sbthlem2 4448	. . . . 5 |- (ran g (_ A -> (A \ (g"(B \ (f"U.D)))) (_ U.D)
4	1, 2	sbthlem1 4447	. . . . 5 |- U.D (_ (A \ (g"(B \ (f"U.D))))
5	3, 4	jctil 292	. . . 4 |- (ran g (_ A -> (U.D (_ (A \ (g"(B \ (f"U.D)))) /\ (A \ (g"(B \ (f"U.D)))) (_ U.D))
6		eqss 2077	. . . 4 |- (U.D = (A \ (g"(B \ (f"U.D)))) <-> (U.D (_ (A \ (g"(B \ (f"U.D)))) /\ (A \ (g"(B \ (f"U.D)))) (_ U.D))
7	5, 6	sylibr 200	. . 3 |- (ran g (_ A -> U.D = (A \ (g"(B \ (f"U.D)))))
8	7	difeq2d 2159	. 2 |- (ran g (_ A -> (A \ U.D) = (A \ (A \ (g"(B \ (f"U.D))))))
9		imassrn 3415	. . . 4 |- (g"(B \ (f"U.D))) (_ ran g
10		sstr2 2071	. . . 4 |- ((g"(B \ (f"U.D))) (_ ran g -> (ran g (_ A -> (g"(B \ (f"U.D))) (_ A))
11	9, 10	ax-mp 7	. . 3 |- (ran g (_ A -> (g"(B \ (f"U.D))) (_ A)
12		dfss4 2242	. . 3 |- ((g"(B \ (f"U.D))) (_ A <-> (A \ (A \ (g"(B \ (f"U.D))))) = (g"(B \ (f"U.D))))
13	11, 12	sylib 198	. 2 |- (ran g (_ A -> (A \ (A \ (g"(B \ (f"U.D))))) = (g"(B \ (f"U.D))))
14	8, 13	eqtr2d 1508	1 |- (ran g (_ A -> (g"(B \ (f"U.D))) = (A \ U.D))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   \ cdif 2044   (_ wss 2047  U.cuni 2503  ran crn 3171  "cima 3173

This theorem is referenced by:  sbthlem4 4450  sbthlem5 4451

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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