Theorem sbthlem3 4438

Description: Lemma for sbth 4446.

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 4437	. . . . 5 |- (ran g (_ A -> (A \ (g"(B \ (f"U.D)))) (_ U.D)
4	1, 2	sbthlem1 4436	. . . . 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 2074	. . . 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 2156	. 2 |- (ran g (_ A -> (A \ U.D) = (A \ (A \ (g"(B \ (f"U.D))))))
9		imassrn 3411	. . . 4 |- (g"(B \ (f"U.D))) (_ ran g
10		sstr2 2068	. . . 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 2239	. . 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 1506	1 |- (ran g (_ A -> (g"(B \ (f"U.D))) = (A \ U.D))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  {cab 1462  Vcvv 1808   \ cdif 2041   (_ wss 2044  U.cuni 2499  ran crn 3167  "cima 3169

This theorem is referenced by:  sbthlem4 4439  sbthlem5 4440

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-rel 3181  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187

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