Theorem mapvalg 7449

Description: The value of set exponentiation. is the set of all functions that map from to . Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
mapvalg

Distinct variable groups:   ,   ,

Proof of Theorem mapvalg

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

Step	Hyp	Ref	Expression
1		mapex 7445	. . 3
2	1	ancoms 453	. 2
3		elex 3118	. . 3
4		elex 3118	. . 3
5		feq3 5720	. . . . . 6
6	5	abbidv 2593	. . . . 5
7		feq2 5719	. . . . . 6
8	7	abbidv 2593	. . . . 5
9		df-map 7441	. . . . 5
10	6, 8, 9	ovmpt2g 6437	. . . 4
11	10	3expia 1198	. . 3
12	3, 4, 11	syl2an 477	. 2
13	2, 12	mpd 15	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  cvv 3109  -->wf 5589  (class class class)co 6296  cmap 7439

This theorem is referenced by:  mapval  7451  elmapg  7452  ixpconstg  7498  hashf1lem2  12505  symgbasfi  16411  birthdaylem1  23281  birthdaylem2  23282  cnfex  31403

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-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-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-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  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-map 7441