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Theorem f1stres 6360
 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 2951 . . . . . . . 8
2 vex 2951 . . . . . . . 8
31, 2op1sta 5343 . . . . . . 7
43eleq1i 2498 . . . . . 6
54biimpri 198 . . . . 5
65adantr 452 . . . 4
76rgen2 2794 . . 3
8 sneq 3817 . . . . . . 7
98dmeqd 5064 . . . . . 6
109unieqd 4018 . . . . 5
1110eleq1d 2501 . . . 4
1211ralxp 5008 . . 3
137, 12mpbir 201 . 2
14 df-1st 6341 . . . . 5
1514reseq1i 5134 . . . 4
16 ssv 3360 . . . . 5
17 resmpt 5183 . . . . 5
1816, 17ax-mp 8 . . . 4
1915, 18eqtri 2455 . . 3
2019fmpt 5882 . 2
2113, 20mpbi 200 1
 Colors of variables: wff set class Syntax hints:   wceq 1652   wcel 1725  wral 2697  cvv 2948   wss 3312  csn 3806  cop 3809  cuni 4007   cmpt 4258   cxp 4868   cdm 4870   cres 4872  wf 5442  c1st 6339 This theorem is referenced by:  fo1stres  6362  1stcof  6366  fparlem1  6438  domssex2  7259  domssex  7260  unxpwdom2  7548  1stfcl  14286  tx1cn  17633  xpinpreima  24296  xpinpreima2  24297  1stmbfm  24602  hausgraph  27499 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-1st 6341
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