Theorem cmpval 10658

Description: Value of the identity function expressed with the 2ndfunctions.

Hypothesis

Ref	Expression
cmpval.1	|- G = (o` T)

Assertion

Ref	Expression
cmpval	|- G = (2nd` (2nd` T))

Proof of Theorem cmpval

Step	Hyp	Ref	Expression
1		cmpval.1	. 2 |- G = (o` T)
2		fo2nd 4092	. . . . . 6 |- 2nd:V-onto->V
3		fofun 3673	. . . . . 6 |- (2nd:V-onto->V -> Fun 2nd)
4	2, 3	ax-mp 7	. . . . 5 |- Fun 2nd
5		fof 3672	. . . . . 6 |- (2nd:V-onto->V -> 2nd:V-->V)
6	2, 5	ax-mp 7	. . . . 5 |- 2nd:V-->V
7		fvco3 3776	. . . . 5 |- ((Fun 2nd /\ 2nd:V-->V /\ T e. V) -> ((2nd o. 2nd)` T) = (2nd` (2nd` T)))
8	4, 6, 7	mp3an12 906	. . . 4 |- (T e. V -> ((2nd o. 2nd)` T) = (2nd` (2nd` T)))
9		df-cmpa 10652	. . . . 5 |- o = (2nd o. 2nd)
10	9	fveq1i 3725	. . . 4 |- (o` T) = ((2nd o. 2nd)` T)
11	8, 10	syl5eq 1519	. . 3 |- (T e. V -> (o` T) = (2nd` (2nd` T)))
12		fvprc 3721	. . . 4 |- (-. T e. V -> (o` T) = (/))
13		fvprc 3721	. . . . . 6 |- (-. T e. V -> (2nd` T) = (/))
14	13	fveq2d 3728	. . . . 5 |- (-. T e. V -> (2nd` (2nd` T)) = (2nd` (/)))
15		2nd0 4084	. . . . 5 |- (2nd` (/)) = (/)
16	14, 15	syl6req 1524	. . . 4 |- (-. T e. V -> (/) = (2nd` (2nd` T)))
17	12, 16	eqtrd 1507	. . 3 |- (-. T e. V -> (o` T) = (2nd` (2nd` T)))
18	11, 17	pm2.61i 126	. 2 |- (o` T) = (2nd` (2nd` T))
19	1, 18	eqtr 1495	1 |- G = (2nd` (2nd` T))

Syntax hints:  -. wn 2   = wceq 956   e. wcel 958  Vcvv 1811  (/)c0 2280   o. ccom 3174  Fun wfun 3176  -->wf 3178  -onto->wfo 3180  ` cfv 3182  2ndc2nd 4078  oco_ 10647

This theorem is referenced by:  algi 10660  dedi 10670  dedalg 10676  cati 10688  catded 10697

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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-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-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-2nd 4080  df-cmpa 10652

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