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Theorem ofrval 5664
Description: Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
offval.1  F  Fn
offval.2  G  Fn
offval.3  V
offval.4  W
offval.5  i^i  S
ofrval.6  X  F `  X  C
ofrval.7  X  G `  X  D
Assertion
Ref Expression
ofrval  F  o R R G  X  S  C R D

Proof of Theorem ofrval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . . . 6  F  Fn
2 offval.2 . . . . . 6  G  Fn
3 offval.3 . . . . . 6  V
4 offval.4 . . . . . 6  W
5 offval.5 . . . . . 6  i^i  S
6 eqidd 2038 . . . . . 6  F `  F `
7 eqidd 2038 . . . . . 6  G `  G `
81, 2, 3, 4, 5, 6, 7ofrfval 5662 . . . . 5  F  o R R G  S  F `  R G `
98biimpa 280 . . . 4  F  o R R G  S  F `  R G `
10 fveq2 5121 . . . . . 6  X  F `  F `  X
11 fveq2 5121 . . . . . 6  X  G `  G `  X
1210, 11breq12d 3768 . . . . 5  X  F `  R G `  F `  X R G `  X
1312rspccv 2647 . . . 4  S  F `  R G `  X  S  F `  X R G `  X
149, 13syl 14 . . 3  F  o R R G  X  S  F `  X R G `  X
15143impia 1100 . 2  F  o R R G  X  S  F `  X R G `  X
16 simp1 903 . . 3  F  o R R G  X  S
17 inss1 3151 . . . . 5  i^i  C_
185, 17eqsstr3i 2970 . . . 4  S  C_
19 simp3 905 . . . 4  F  o R R G  X  S  X  S
2018, 19sseldi 2937 . . 3  F  o R R G  X  S  X
21 ofrval.6 . . 3  X  F `  X  C
2216, 20, 21syl2anc 391 . 2  F  o R R G  X  S  F `  X  C
23 inss2 3152 . . . . 5  i^i  C_
245, 23eqsstr3i 2970 . . . 4  S  C_
2524, 19sseldi 2937 . . 3  F  o R R G  X  S  X
26 ofrval.7 . . 3  X  G `  X  D
2716, 25, 26syl2anc 391 . 2  F  o R R G  X  S  G `  X  D
2815, 22, 273brtr3d 3784 1  F  o R R G  X  S  C R D
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390  wral 2300    i^i cin 2910   class class class wbr 3755    Fn wfn 4840   ` cfv 4845    o Rcofr 5653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ofr 5655
This theorem is referenced by: (None)
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