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Theorem f1stres 6822
Description: Mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
f1stres

Proof of Theorem f1stres
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . . . . . 8
2 vex 3112 . . . . . . . 8
31, 2op1sta 5495 . . . . . . 7
43eleq1i 2534 . . . . . 6
54biimpri 206 . . . . 5
65adantr 465 . . . 4
76rgen2 2882 . . 3
8 sneq 4039 . . . . . . 7
98dmeqd 5210 . . . . . 6
109unieqd 4259 . . . . 5
1110eleq1d 2526 . . . 4
1211ralxp 5149 . . 3
137, 12mpbir 209 . 2
14 df-1st 6800 . . . . 5
1514reseq1i 5274 . . . 4
16 ssv 3523 . . . . 5
17 resmpt 5328 . . . . 5
1816, 17ax-mp 5 . . . 4
1915, 18eqtri 2486 . . 3
2019fmpt 6052 . 2
2113, 20mpbi 208 1
Colors of variables: wff setvar class
Syntax hints:  =wceq 1395  e.wcel 1818  A.wral 2807   cvv 3109  C_wss 3475  {csn 4029  <.cop 4035  U.cuni 4249  e.cmpt 4510  X.cxp 5002  domcdm 5004  |`cres 5006  -->wf 5589   c1st 6798
This theorem is referenced by:  fo1stres  6824  1stcof  6828  fparlem1  6900  domssex2  7697  domssex  7698  unxpwdom2  8035  1stfcl  15466  tx1cn  20110  xpinpreima  27888  xpinpreima2  27889  1stmbfm  28231  hausgraph  31172
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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
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-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-fv 5601  df-1st 6800
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