Theorem idval 10657

Description: Value of the identity function expressed with the 1st and 2nd functions.

Hypothesis

Ref	Expression
idval.1	|- J = (id` T)

Assertion

Ref	Expression
idval	|- J = (1st` (2nd` T))

Proof of Theorem idval

Step	Hyp	Ref	Expression
1		idval.1	. 2 |- J = (id` T)
2		fo1st 4091	. . . . . 6 |- 1st:V-onto->V
3		fofun 3673	. . . . . 6 |- (1st:V-onto->V -> Fun 1st)
4	2, 3	ax-mp 7	. . . . 5 |- Fun 1st
5		fo2nd 4092	. . . . . 6 |- 2nd:V-onto->V
6		fof 3672	. . . . . 6 |- (2nd:V-onto->V -> 2nd:V-->V)
7	5, 6	ax-mp 7	. . . . 5 |- 2nd:V-->V
8		fvco3 3776	. . . . 5 |- ((Fun 1st /\ 2nd:V-->V /\ T e. V) -> ((1st o. 2nd)` T) = (1st` (2nd` T)))
9	4, 7, 8	mp3an12 906	. . . 4 |- (T e. V -> ((1st o. 2nd)` T) = (1st` (2nd` T)))
10		df-ida 10651	. . . . 5 |- id = (1st o. 2nd)
11	10	fveq1i 3725	. . . 4 |- (id` T) = ((1st o. 2nd)` T)
12	9, 11	syl5eq 1519	. . 3 |- (T e. V -> (id` T) = (1st` (2nd` T)))
13		fvprc 3721	. . . 4 |- (-. T e. V -> (id` T) = (/))
14		fvprc 3721	. . . . . 6 |- (-. T e. V -> (2nd` T) = (/))
15	14	fveq2d 3728	. . . . 5 |- (-. T e. V -> (1st` (2nd` T)) = (1st` (/)))
16		1st0 4083	. . . . 5 |- (1st` (/)) = (/)
17	15, 16	syl6req 1524	. . . 4 |- (-. T e. V -> (/) = (1st` (2nd` T)))
18	13, 17	eqtrd 1507	. . 3 |- (-. T e. V -> (id` T) = (1st` (2nd` T)))
19	12, 18	pm2.61i 126	. 2 |- (id` T) = (1st` (2nd` T))
20	1, 19	eqtr 1495	1 |- J = (1st` (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  1stc1st 4077  2ndc2nd 4078  idcid_ 10646

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-1st 4079  df-2nd 4080  df-ida 10651

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