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Theorem fliftf 6029
 Description: The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1
flift.2
flift.3
Assertion
Ref Expression
fliftf
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem fliftf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . . . 5
2 flift.1 . . . . . . . . . . 11
3 flift.2 . . . . . . . . . . 11
4 flift.3 . . . . . . . . . . 11
52, 3, 4fliftel 6023 . . . . . . . . . 10
65exbidv 1636 . . . . . . . . 9
76adantr 452 . . . . . . . 8
8 rexcom4 2967 . . . . . . . . 9
9 elisset 2958 . . . . . . . . . . . . . 14
104, 9syl 16 . . . . . . . . . . . . 13
1110biantrud 494 . . . . . . . . . . . 12
12 19.42v 1928 . . . . . . . . . . . 12
1311, 12syl6rbbr 256 . . . . . . . . . . 11
1413rexbidva 2714 . . . . . . . . . 10
1514adantr 452 . . . . . . . . 9
168, 15syl5bbr 251 . . . . . . . 8
177, 16bitrd 245 . . . . . . 7
1817abbidv 2549 . . . . . 6
19 df-dm 4880 . . . . . 6
20 eqid 2435 . . . . . . 7
2120rnmpt 5108 . . . . . 6
2218, 19, 213eqtr4g 2492 . . . . 5
23 df-fn 5449 . . . . 5
241, 22, 23sylanbrc 646 . . . 4
252, 3, 4fliftrel 6022 . . . . . . 7
2625adantr 452 . . . . . 6
27 rnss 5090 . . . . . 6
2826, 27syl 16 . . . . 5
29 rnxpss 5293 . . . . 5
3028, 29syl6ss 3352 . . . 4
31 df-f 5450 . . . 4
3224, 30, 31sylanbrc 646 . . 3
3332ex 424 . 2
34 ffun 5585 . 2
3533, 34impbid1 195 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725  cab 2421  wrex 2698   wss 3312  cop 3809   class class class wbr 4204   cmpt 4258   cxp 4868   cdm 4870   crn 4871   wfun 5440   wfn 5441  wf 5442 This theorem is referenced by:  qliftf  6984  cygznlem2a  16840  pi1xfrf  19070  pi1cof  19076 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-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-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
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