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Theorem fliftf 6226
 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 468 . . . . 5
2 flift.1 . . . . . . . . . . 11
3 flift.2 . . . . . . . . . . 11
4 flift.3 . . . . . . . . . . 11
52, 3, 4fliftel 6220 . . . . . . . . . 10
65exbidv 1776 . . . . . . . . 9
76adantr 472 . . . . . . . 8
8 rexcom4 3053 . . . . . . . . 9
9 elisset 3043 . . . . . . . . . . . . . 14
104, 9syl 17 . . . . . . . . . . . . 13
1110biantrud 515 . . . . . . . . . . . 12
12 19.42v 1842 . . . . . . . . . . . 12
1311, 12syl6rbbr 272 . . . . . . . . . . 11
1413rexbidva 2889 . . . . . . . . . 10
1514adantr 472 . . . . . . . . 9
168, 15syl5bbr 267 . . . . . . . 8
177, 16bitrd 261 . . . . . . 7
1817abbidv 2589 . . . . . 6
19 df-dm 4849 . . . . . 6
20 eqid 2471 . . . . . . 7
2120rnmpt 5086 . . . . . 6
2218, 19, 213eqtr4g 2530 . . . . 5
23 df-fn 5592 . . . . 5
241, 22, 23sylanbrc 677 . . . 4
252, 3, 4fliftrel 6219 . . . . . . 7
2625adantr 472 . . . . . 6
27 rnss 5069 . . . . . 6
2826, 27syl 17 . . . . 5
29 rnxpss 5275 . . . . 5
3028, 29syl6ss 3430 . . . 4
31 df-f 5593 . . . 4
3224, 30, 31sylanbrc 677 . . 3
3332ex 441 . 2
34 ffun 5742 . 2
3533, 34impbid1 208 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452  wex 1671   wcel 1904  cab 2457  wrex 2757   wss 3390  cop 3965   class class class wbr 4395   cmpt 4454   cxp 4837   cdm 4839   crn 4840   wfun 5583   wfn 5584  wf 5585 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597 This theorem is referenced by:  qliftf  7469  cygznlem2a  19215  pi1xfrf  22162  pi1cof  22168
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