Theorem cmpval 10609

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 4089	. . . . . 6 |- 2nd:V-onto->V
3		fofun 3670	. . . . . 6 |- (2nd:V-onto->V -> Fun 2nd)
4	2, 3	ax-mp 7	. . . . 5 |- Fun 2nd
5		fof 3669	. . . . . 6 |- (2nd:V-onto->V -> 2nd:V-->V)
6	2, 5	ax-mp 7	. . . . 5 |- 2nd:V-->V
7		fvco3 3773	. . . . 5 |- ((Fun 2nd /\ 2nd:V-->V /\ T e. V) -> ((2nd o. 2nd)` T) = (2nd` (2nd` T)))
8	4, 6, 7	mp3an12 905	. . . 4 |- (T e. V -> ((2nd o. 2nd)` T) = (2nd` (2nd` T)))
9		df-cmpa 10603	. . . . 5 |- o = (2nd o. 2nd)
10	9	fveq1i 3722	. . . 4 |- (o` T) = ((2nd o. 2nd)` T)
11	8, 10	syl5eq 1518	. . 3 |- (T e. V -> (o` T) = (2nd` (2nd` T)))
12		fvprc 3718	. . . 4 |- (-. T e. V -> (o` T) = (/))
13		fvprc 3718	. . . . . 6 |- (-. T e. V -> (2nd` T) = (/))
14	13	fveq2d 3725	. . . . 5 |- (-. T e. V -> (2nd` (2nd` T)) = (2nd` (/)))
15		2nd0 4081	. . . . 5 |- (2nd` (/)) = (/)
16	14, 15	syl6req 1523	. . . 4 |- (-. T e. V -> (/) = (2nd` (2nd` T)))
17	12, 16	eqtrd 1506	. . 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 1494	1 |- G = (2nd` (2nd` T))

Syntax hints:  -. wn 2   = wceq 955   e. wcel 957  Vcvv 1809  (/)c0 2278   o. ccom 3171  Fun wfun 3173  -->wf 3175  -onto->wfo 3177  ` cfv 3179  2ndc2nd 4075  oco_ 10598

This theorem is referenced by:  algi 10611  dedi 10621  dedalg 10627  cati 10639  catded 10648

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-9 964  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 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fo 3193  df-fv 3195  df-2nd 4077  df-cmpa 10603

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