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Theorem fvray 26080
 Description: Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvray Ray OutsideOf
Distinct variable groups:   ,   ,   ,

Proof of Theorem fvray
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 6087 . 2 Ray Ray
2 eqid 2438 . . . . 5 OutsideOf OutsideOf
3 fveq2 5731 . . . . . . . . 9
43eleq2d 2505 . . . . . . . 8
53eleq2d 2505 . . . . . . . 8
64, 53anbi12d 1256 . . . . . . 7
7 rabeq 2952 . . . . . . . . 9 OutsideOf OutsideOf
83, 7syl 16 . . . . . . . 8 OutsideOf OutsideOf
98eqeq2d 2449 . . . . . . 7 OutsideOf OutsideOf OutsideOf OutsideOf
106, 9anbi12d 693 . . . . . 6 OutsideOf OutsideOf OutsideOf OutsideOf
1110rspcev 3054 . . . . 5 OutsideOf OutsideOf OutsideOf OutsideOf
122, 11mpanr2 667 . . . 4 OutsideOf OutsideOf
13 simpr1 964 . . . . 5
14 simpr2 965 . . . . 5
15 fvex 5745 . . . . . . 7
1615rabex 4357 . . . . . 6 OutsideOf
17 eleq1 2498 . . . . . . . . . 10
18 neeq1 2611 . . . . . . . . . 10
1917, 183anbi13d 1257 . . . . . . . . 9
20 breq1 4218 . . . . . . . . . . 11 OutsideOf OutsideOf
2120rabbidv 2950 . . . . . . . . . 10 OutsideOf OutsideOf
2221eqeq2d 2449 . . . . . . . . 9 OutsideOf OutsideOf
2319, 22anbi12d 693 . . . . . . . 8 OutsideOf OutsideOf
2423rexbidv 2728 . . . . . . 7 OutsideOf OutsideOf
25 eleq1 2498 . . . . . . . . . 10
26 neeq2 2612 . . . . . . . . . 10
2725, 263anbi23d 1258 . . . . . . . . 9
28 opeq1 3986 . . . . . . . . . . . 12
2928breq2d 4227 . . . . . . . . . . 11 OutsideOf OutsideOf
3029rabbidv 2950 . . . . . . . . . 10 OutsideOf OutsideOf
3130eqeq2d 2449 . . . . . . . . 9 OutsideOf OutsideOf
3227, 31anbi12d 693 . . . . . . . 8 OutsideOf OutsideOf
3332rexbidv 2728 . . . . . . 7 OutsideOf OutsideOf
34 eqeq1 2444 . . . . . . . . 9 OutsideOf OutsideOf OutsideOf OutsideOf
3534anbi2d 686 . . . . . . . 8 OutsideOf OutsideOf OutsideOf OutsideOf
3635rexbidv 2728 . . . . . . 7 OutsideOf OutsideOf OutsideOf OutsideOf
3724, 33, 36eloprabg 6164 . . . . . 6 OutsideOf OutsideOf OutsideOf OutsideOf OutsideOf
3816, 37mp3an3 1269 . . . . 5 OutsideOf OutsideOf OutsideOf OutsideOf
3913, 14, 38syl2anc 644 . . . 4 OutsideOf OutsideOf OutsideOf OutsideOf
4012, 39mpbird 225 . . 3 OutsideOf OutsideOf
41 df-br 4216 . . . . 5 Ray OutsideOf OutsideOf Ray
42 df-ray 26077 . . . . . 6 Ray OutsideOf
4342eleq2i 2502 . . . . 5 OutsideOf Ray OutsideOf OutsideOf
4441, 43bitri 242 . . . 4 Ray OutsideOf OutsideOf OutsideOf
45 funray 26079 . . . . 5 Ray
46 funbrfv 5768 . . . . 5 Ray Ray OutsideOf Ray OutsideOf
4745, 46ax-mp 5 . . . 4 Ray OutsideOf Ray OutsideOf
4844, 47sylbir 206 . . 3 OutsideOf OutsideOf Ray OutsideOf
4940, 48syl 16 . 2 Ray OutsideOf
501, 49syl5eq 2482 1 Ray OutsideOf
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726   wne 2601  wrex 2708  crab 2711  cvv 2958  cop 3819   class class class wbr 4215   wfun 5451  cfv 5457  (class class class)co 6084  coprab 6085  cn 10005  cee 25832  OutsideOfcoutsideof 26058  Raycray 26074 This theorem is referenced by:  lineunray  26086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-z 10288  df-uz 10494  df-fz 11049  df-ee 25835  df-ray 26077
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