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Theorem bnj1280 33556
 Description: Technical lemma for bnj60 33598. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1280.1
bnj1280.2
bnj1280.3
bnj1280.4
bnj1280.5
bnj1280.6
bnj1280.7
bnj1280.17
Assertion
Ref Expression
bnj1280
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,   ,   ,,   ,   ,   ,   ,,   ,,
Allowed substitution hints:   (,,,,,)   (,,,,,)   (,,,)   (,,)   (,,,,,)   (,,,)   (,,,)   (,,,,,)   (,,)   (,,,)

Proof of Theorem bnj1280
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bnj1280.1 . . . . . . . 8
2 bnj1280.2 . . . . . . . 8
3 bnj1280.3 . . . . . . . 8
4 bnj1280.4 . . . . . . . 8
5 bnj1280.5 . . . . . . . 8
6 bnj1280.6 . . . . . . . 8
7 bnj1280.7 . . . . . . . 8
81, 2, 3, 4, 5, 6, 7bnj1286 33555 . . . . . . 7
98sseld 3508 . . . . . 6
10 bnj1280.17 . . . . . . . . 9
11 disj1 3874 . . . . . . . . 9
1210, 11sylib 196 . . . . . . . 8
131219.21bi 1818 . . . . . . 7
14 fveq2 5872 . . . . . . . . . . 11
15 fveq2 5872 . . . . . . . . . . 11
1614, 15neeq12d 2746 . . . . . . . . . 10
1716, 5elrab2 3268 . . . . . . . . 9
1817notbii 296 . . . . . . . 8
19 imnan 422 . . . . . . . 8
20 nne 2668 . . . . . . . . 9
2120imbi2i 312 . . . . . . . 8
2218, 19, 213bitr2i 273 . . . . . . 7
2313, 22syl6ib 226 . . . . . 6
249, 23mpdd 40 . . . . 5
2524imp 429 . . . 4
26 fvres 5886 . . . . . 6
279, 26syl6 33 . . . . 5
2827imp 429 . . . 4
29 fvres 5886 . . . . . 6
309, 29syl6 33 . . . . 5
3130imp 429 . . . 4
3225, 28, 313eqtr4d 2518 . . 3
3332ralrimiva 2881 . 2
34 resabs1 5308 . . . . 5
358, 34syl 16 . . . 4
36 resabs1 5308 . . . . 5
378, 36syl 16 . . . 4
3835, 37eqeq12d 2489 . . 3
391, 2, 3, 4, 5, 6, 7bnj1256 33551 . . . . . . 7
404bnj1292 33354 . . . . . . . . 9
41 fndm 5686 . . . . . . . . 9
4240, 41syl5sseq 3557 . . . . . . . 8
43 fnssres 5700 . . . . . . . 8
4442, 43mpdan 668 . . . . . . 7
4539, 44bnj31 33253 . . . . . 6
4645bnj1265 33351 . . . . 5
477, 46bnj835 33297 . . . 4
481, 2, 3, 4, 5, 6, 7bnj1259 33552 . . . . . . 7
494bnj1293 33355 . . . . . . . . 9
50 fndm 5686 . . . . . . . . 9
5149, 50syl5sseq 3557 . . . . . . . 8
52 fnssres 5700 . . . . . . . 8
5351, 52mpdan 668 . . . . . . 7
5448, 53bnj31 33253 . . . . . 6
5554bnj1265 33351 . . . . 5
567, 55bnj835 33297 . . . 4
57 fvreseq 5990 . . . 4
5847, 56, 8, 57syl21anc 1227 . . 3
5938, 58bitr3d 255 . 2
6033, 59mpbird 232 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369   w3a 973  wal 1377   wceq 1379   wcel 1767  cab 2452   wne 2662  wral 2817  wrex 2818  crab 2821   cin 3480   wss 3481  c0 3790  cop 4039   class class class wbr 4453   cdm 5005   cres 5007   wfn 5589  cfv 5594   w-bnj17 33219   c-bnj14 33221   w-bnj15 33225 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-bnj17 33220 This theorem is referenced by:  bnj1311  33560
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