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Theorem funray 26076
 Description: Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funray Ray

Proof of Theorem funray
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeanv 2877 . . . . . 6 OutsideOf OutsideOf OutsideOf OutsideOf
2 simp1 958 . . . . . . . . . . 11
3 simp1 958 . . . . . . . . . . 11
4 axdimuniq 25854 . . . . . . . . . . . . . . 15
5 fveq2 5730 . . . . . . . . . . . . . . . . . . 19
6 rabeq 2952 . . . . . . . . . . . . . . . . . . 19 OutsideOf OutsideOf
75, 6syl 16 . . . . . . . . . . . . . . . . . 18 OutsideOf OutsideOf
87eqeq2d 2449 . . . . . . . . . . . . . . . . 17 OutsideOf OutsideOf
98anbi1d 687 . . . . . . . . . . . . . . . 16 OutsideOf OutsideOf OutsideOf OutsideOf
10 eqtr3 2457 . . . . . . . . . . . . . . . 16 OutsideOf OutsideOf
119, 10syl6bi 221 . . . . . . . . . . . . . . 15 OutsideOf OutsideOf
124, 11syl 16 . . . . . . . . . . . . . 14 OutsideOf OutsideOf
1312an4s 801 . . . . . . . . . . . . 13 OutsideOf OutsideOf
1413ex 425 . . . . . . . . . . . 12 OutsideOf OutsideOf
1514com3l 78 . . . . . . . . . . 11 OutsideOf OutsideOf
162, 3, 15syl2an 465 . . . . . . . . . 10 OutsideOf OutsideOf
1716imp 420 . . . . . . . . 9 OutsideOf OutsideOf
1817an4s 801 . . . . . . . 8 OutsideOf OutsideOf
1918com12 30 . . . . . . 7 OutsideOf OutsideOf
2019rexlimivv 2837 . . . . . 6 OutsideOf OutsideOf
211, 20sylbir 206 . . . . 5 OutsideOf OutsideOf
2221gen2 1557 . . . 4 OutsideOf OutsideOf
23 eqeq1 2444 . . . . . . . 8 OutsideOf OutsideOf
2423anbi2d 686 . . . . . . 7 OutsideOf OutsideOf
2524rexbidv 2728 . . . . . 6 OutsideOf OutsideOf
265eleq2d 2505 . . . . . . . . 9
275eleq2d 2505 . . . . . . . . 9
2826, 273anbi12d 1256 . . . . . . . 8
297eqeq2d 2449 . . . . . . . 8 OutsideOf OutsideOf
3028, 29anbi12d 693 . . . . . . 7 OutsideOf OutsideOf
3130cbvrexv 2935 . . . . . 6 OutsideOf OutsideOf
3225, 31syl6bb 254 . . . . 5 OutsideOf OutsideOf
3332mo4 2316 . . . 4 OutsideOf OutsideOf OutsideOf
3422, 33mpbir 202 . . 3 OutsideOf
3534funoprab 6172 . 2 OutsideOf
36 df-ray 26074 . . 3 Ray OutsideOf
3736funeqi 5476 . 2 Ray OutsideOf
3835, 37mpbir 202 1 Ray
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937  wal 1550   wceq 1653   wcel 1726  wmo 2284   wne 2601  wrex 2708  crab 2711  cop 3819   class class class wbr 4214   wfun 5450  cfv 5456  coprab 6084  cn 10002  cee 25829  OutsideOfcoutsideof 26055  Raycray 26071 This theorem is referenced by:  fvray  26077 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069 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 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-z 10285  df-uz 10491  df-fz 11046  df-ee 25832  df-ray 26074
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