Theorem metcnp4lem1 7965

Description: Lemma for metcnp4 7967.

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 3730	. . 3 |- (j = k -> (f` j) = (f` k))
2	1	fveq2d 3734	. 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 3738	. 2 |- (F` (f` k)) e. V
5	2, 3, 4	fvopab4 3786	1 |- (k e. NN -> (G` k) = (F` (f` k)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {copab 2671  dom cdm 3176  ` cfv 3188  NNcn 5308  Opencopn 7789

This theorem is referenced by:  metcnp4lem2 7966  metcnp4 7967

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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