Theorem ofmres 6796

Description: Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, (oF|`(AX.)) can be a set by ofmresex 6797, allowing it to be used as a function or structure argument. By ofmresval 6552, the restricted operation map values are the same as the original values, allowing theorems for to be reused. (Contributed by NM, 20-Oct-2014.)

Assertion

Ref	Expression
ofmres

Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem ofmres

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3523	. . 3
2		ssv 3523	. . 3
3		resmpt2 6400	. . 3
4	1, 2, 3	mp2an 672	. 2
5		df-of 6540	. . 3
6	5	reseq1i 5274	. 2
7		eqid 2457	. . 3
8		eqid 2457	. . 3
9		vex 3112	. . . 4
10		vex 3112	. . . 4
11	9	dmex 6733	. . . . . 6
12	11	inex1 4593	. . . . 5
13	12	mptex 6143	. . . 4
14	5	ovmpt4g 6425	. . . 4
15	9, 10, 13, 14	mp3an 1324	. . 3
16	7, 8, 15	mpt2eq123i 6360	. 2
17	4, 6, 16	3eqtr4i 2496	1

Syntax hints:  =wceq 1395  e.wcel 1818  cvv 3109  i^icin 3474  C_wss 3475  e.cmpt 4510  X.cxp 5002  domcdm 5004  |`cres 5006  `cfv 5593  (class class class)co 6296  e.cmpt2 6298  oFcof 6538

This theorem is referenced by:  mplsubrglem  18100  mplsubrglemOLD  18101  psrplusgpropd  18277

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-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-reu 2814  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-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-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-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6540