Theorem metcnp4lem1 7968

Description: Lemma for metcnp4 7970.

Hypotheses

Ref	Expression
metcnp4.1	|- X = dom dom C
metcnp4.3	|- Y = dom dom D
metcnp4.c	|- J = (Open` C)
metcnp4.d	|- K = (Open` D)
metcnp4.5	|- G = {<.j, y>. | (j e. NN /\ y = (F` (f` j)))}

Assertion

Ref	Expression
metcnp4lem1	|- (k e. NN -> (G` k) = (F` (f` k)))

Distinct variable groups:   f,k,C   D,f,k   f,j,y,F,k   k,G   f,X,j,k   f,Y,j,k,y

Proof of Theorem metcnp4lem1

Step	Hyp	Ref	Expression
1		fveq2 3724	. . 3 |- (j = k -> (f` j) = (f` k))
2	1	fveq2d 3728	. 2 |- (j = k -> (F` (f` j)) = (F` (f` k)))
3		metcnp4.5	. 2 |- G = {<.j, y>. | (j e. NN /\ y = (F` (f` j)))}
4		fvex 3732	. 2 |- (F` (f` k)) e. V
5	2, 3, 4	fvopab4 3780	1 |- (k e. NN -> (G` k) = (F` (f` k)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {copab 2666  dom cdm 3170  ` cfv 3182  NNcn 5296  Opencopn 7792

This theorem is referenced by:  metcnp4lem2 7969  metcnp4 7970

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-9 965  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  ax-un 2866

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-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  df-fv 3198

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