Theorem idval 10537

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 4081	. . . . . 6 |- 1st:V-onto->V
3		fofun 3664	. . . . . 6 |- (1st:V-onto->V -> Fun 1st)
4	2, 3	ax-mp 7	. . . . 5 |- Fun 1st
5		fo2nd 4082	. . . . . 6 |- 2nd:V-onto->V
6		fof 3663	. . . . . 6 |- (2nd:V-onto->V -> 2nd:V-->V)
7	5, 6	ax-mp 7	. . . . 5 |- 2nd:V-->V
8		fvco3 3767	. . . . 5 |- ((Fun 1st /\ 2nd:V-->V /\ T e. V) -> ((1st o. 2nd)` T) = (1st` (2nd` T)))
9	4, 7, 8	mp3an12 904	. . . 4 |- (T e. V -> ((1st o. 2nd)` T) = (1st` (2nd` T)))
10		df-ida 10531	. . . . 5 |- id = (1st o. 2nd)
11	10	fveq1i 3716	. . . 4 |- (id` T) = ((1st o. 2nd)` T)
12	9, 11	syl5eq 1516	. . 3 |- (T e. V -> (id` T) = (1st` (2nd` T)))
13		fvprc 3712	. . . 4 |- (-. T e. V -> (id` T) = (/))
14		fvprc 3712	. . . . . 6 |- (-. T e. V -> (2nd` T) = (/))
15	14	fveq2d 3719	. . . . 5 |- (-. T e. V -> (1st` (2nd` T)) = (1st` (/)))
16		1st0 4073	. . . . 5 |- (1st` (/)) = (/)
17	15, 16	syl6req 1521	. . . 4 |- (-. T e. V -> (/) = (1st` (2nd` T)))
18	13, 17	eqtrd 1504	. . 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 1492	1 |- J = (1st` (2nd` T))

Syntax hints:  -. wn 2   = wceq 954   e. wcel 956  Vcvv 1807  (/)c0 2276   o. ccom 3169  Fun wfun 3171  -->wf 3173  -onto->wfo 3175  ` cfv 3177  1stc1st 4067  2ndc2nd 4068  idcid_ 10526

This theorem is referenced by:  algi 10540  dedi 10550  dedalg 10556  cati 10568  catded 10577

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193  df-1st 4069  df-2nd 4070  df-ida 10531

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