Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prmapcp2 Unicode version

Theorem prmapcp2 25260
 Description: A projection is a mapping from a cartesian product to an element of the family implied in the product. Bourbaki E.II.34 cor. 1. (Contributed by FL, 19-Jun-2011.)
Hypotheses
Ref Expression
prmapcp2.1
prmapcp2.2
Assertion
Ref Expression
prmapcp2
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem prmapcp2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 prmapcp2.1 . . . . . 6
21eleq2i 2360 . . . . 5
32biimpi 186 . . . 4
4 simpr 447 . . . 4
5 prmapcp2.2 . . . . 5
65bclelnu 25258 . . . 4
73, 4, 6syl2anr 464 . . 3
8 eqid 2296 . . 3
97, 8fmptd 5700 . 2
10 ispr1 25259 . . 3
1110feq1d 5395 . 2
129, 11mpbird 223 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1632   wcel 1696   cmpt 4093  wf 5267  cfv 5271  (class class class)co 5874  cixp 6833   cpro 25253 This theorem is referenced by:  usptoplem  25649  istopx  25650  prcnt  25654 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-ixp 6834  df-pro 25255
 Copyright terms: Public domain W3C validator