Theorem domval 10499

Description: Value of the domain function expressed with the 1st function.

Hypothesis

Ref	Expression
domval.1	|- D = (dom` T)

Assertion

Ref	Expression
domval	|- D = (1st` (1st` T))

Proof of Theorem domval

Step	Hyp	Ref	Expression
1		domval.1	. 2 |- D = (dom` T)
2		fo1st 4075	. . . . . 6 |- 1st:V-onto->V
3		fofun 3658	. . . . . 6 |- (1st:V-onto->V -> Fun 1st)
4	2, 3	ax-mp 7	. . . . 5 |- Fun 1st
5		fof 3657	. . . . . 6 |- (1st:V-onto->V -> 1st:V-->V)
6	2, 5	ax-mp 7	. . . . 5 |- 1st:V-->V
7		fvco3 3761	. . . . 5 |- ((Fun 1st /\ 1st:V-->V /\ T e. V) -> ((1st o. 1st)` T) = (1st` (1st` T)))
8	4, 6, 7	mp3an12 903	. . . 4 |- (T e. V -> ((1st o. 1st)` T) = (1st` (1st` T)))
9		df-doma 10493	. . . . 5 |- dom = (1st o. 1st)
10	9	fveq1i 3710	. . . 4 |- (dom` T) = ((1st o. 1st)` T)
11	8, 10	syl5eq 1511	. . 3 |- (T e. V -> (dom` T) = (1st` (1st` T)))
12		fvprc 3706	. . . 4 |- (-. T e. V -> (dom` T) = (/))
13		fvprc 3706	. . . . . 6 |- (-. T e. V -> (1st` T) = (/))
14	13	fveq2d 3713	. . . . 5 |- (-. T e. V -> (1st` (1st` T)) = (1st` (/)))
15		1st0 4067	. . . . 5 |- (1st` (/)) = (/)
16	14, 15	syl6req 1516	. . . 4 |- (-. T e. V -> (/) = (1st` (1st` T)))
17	12, 16	eqtrd 1499	. . 3 |- (-. T e. V -> (dom` T) = (1st` (1st` T)))
18	11, 17	pm2.61i 126	. 2 |- (dom` T) = (1st` (1st` T))
19	1, 18	eqtr 1487	1 |- D = (1st` (1st` T))

Syntax hints:  -. wn 2   = wceq 953   e. wcel 955  Vcvv 1802  (/)c0 2270   o. ccom 3164  Fun wfun 3166  -->wf 3168  -onto->wfo 3170  ` cfv 3172  1stc1st 4061  domcdom_ 10488

This theorem is referenced by:  algi 10504  dedi 10514  dedalg 10520  cati 10532  catded 10541

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fo 3186  df-fv 3188  df-1st 4063  df-doma 10493

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