Theorem wemappo 7995

Description: Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values.

Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis

Ref	Expression
wemapso.t

Assertion

Ref	Expression
wemappo

Distinct variable groups:   ,   ,,,,   ,,,,   ,S,,,

Proof of Theorem wemappo

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		simpll3 1037	. . . . . . 7
3		elmapi 7460	. . . . . . . . 9
4	3	adantl 466	. . . . . . . 8
5	4	ffvelrnda 6031	. . . . . . 7
6		poirr 4816	. . . . . . 7
7	2, 5, 6	syl2anc 661	. . . . . 6
8	7	intnanrd 917	. . . . 5
9	8	nrexdv 2913	. . . 4
10		vex 3112	. . . . 5
11		wemapso.t	. . . . . 6
12	11	wemaplem1 7992	. . . . 5
13	10, 10, 12	mp2an 672	. . . 4
14	9, 13	sylnibr 305	. . 3
15		simpll1 1035	. . . . 5
16		simplr1 1038	. . . . 5
17		simplr2 1039	. . . . 5
18		simplr3 1040	. . . . 5
19		simpll2 1036	. . . . 5
20		simpll3 1037	. . . . 5
21		simprl 756	. . . . 5
22		simprr 757	. . . . 5
23	11, 15, 16, 17, 18, 19, 20, 21, 22	wemaplem3 7994	. . . 4
24	23	ex 434	. . 3
25	14, 24	ispod 4813	. 2
26	1, 25	syl3an1 1261	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  cvv 3109   class class class wbr 4452  {copab 4509  Powpo 4803  Orwor 4804  -->wf 5589  `cfv 5593  (class class class)co 6296  cmap 7439

This theorem is referenced by:  wemapsolem  7996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-po 4805  df-so 4806  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801  df-map 7441