Theorem hstrlem2 10181

Description: Lemma for strong set of CH states theorem.

Hypothesis

Ref	Expression
hstrlem2.1	|- S = {<.x, y>. | (x e. CH /\ y = ((proj` x)` u))}

Assertion

Ref	Expression
hstrlem2	|- (C e. CH -> (S` C) = ((proj` C)` u))

Distinct variable groups:   x,y,C   x,u,y

Proof of Theorem hstrlem2

Step	Hyp	Ref	Expression
1		fveq2 3730	. . 3 |- (x = C -> (proj` x) = (proj` C))
2	1	fveq1d 3732	. 2 |- (x = C -> ((proj` x)` u) = ((proj` C)` u))
3		hstrlem2.1	. 2 |- S = {<.x, y>. | (x e. CH /\ y = ((proj` x)` u))}
4		fvex 3738	. 2 |- ((proj` C)` u) e. V
5	2, 3, 4	fvopab4 3786	1 |- (C e. CH -> (S` C) = ((proj` C)` u))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {copab 2671  ` cfv 3188  CHcch 8793  projcpj 8801

This theorem is referenced by:  hstrlem3a 10182  hstrlem4 10184  hstrlem5 10185

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204

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