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Theorem df1stres 23258
 Description: Definition for a restriction of the (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
Assertion
Ref Expression
df1stres
Distinct variable groups:   ,,   ,,

Proof of Theorem df1stres
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df1st2 6221 . . . 4
21reseq1i 4967 . . 3
3 resoprab 5956 . . 3
4 resres 4984 . . . 4
5 incom 3374 . . . . . 6
6 ssv 3211 . . . . . . . 8
7 ssv 3211 . . . . . . . 8
8 xpss12 4808 . . . . . . . 8
96, 7, 8mp2an 653 . . . . . . 7
10 df-ss 3179 . . . . . . 7
119, 10mpbi 199 . . . . . 6
125, 11eqtr3i 2318 . . . . 5
1312reseq2i 4968 . . . 4
144, 13eqtri 2316 . . 3
152, 3, 143eqtr3ri 2325 . 2
16 df-mpt2 5879 . 2
1715, 16eqtr4i 2319 1
 Colors of variables: wff set class Syntax hints:   wa 358   wceq 1632   wcel 1696  cvv 2801   cin 3164   wss 3165   cxp 4703   cres 4707  coprab 5875   cmpt2 5876  c1st 6136 This theorem is referenced by:  cnre2csqima  23310 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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 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-rab 2565  df-v 2803  df-sbc 3005  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-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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139
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