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Theorem bnj1442 33540
 Description: Technical lemma for bnj60 33553. 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
bnj1442.1
bnj1442.2
bnj1442.3
bnj1442.4
bnj1442.5
bnj1442.6
bnj1442.7
bnj1442.8
bnj1442.9
bnj1442.10
bnj1442.11
bnj1442.12
bnj1442.13
bnj1442.14
bnj1442.15
bnj1442.16
bnj1442.17
bnj1442.18
Assertion
Ref Expression
bnj1442
Distinct variable group:   ,
Allowed substitution hints:   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)

Proof of Theorem bnj1442
StepHypRef Expression
1 bnj1442.18 . . 3
2 bnj1442.17 . . . 4
3 bnj1442.16 . . . . . 6
43bnj930 33263 . . . . 5
5 opex 4717 . . . . . . . 8
65snid 4061 . . . . . . 7
7 elun2 3677 . . . . . . 7
86, 7ax-mp 5 . . . . . 6
9 bnj1442.12 . . . . . 6
108, 9eleqtrri 2554 . . . . 5
11 funopfv 5913 . . . . 5
124, 10, 11mpisyl 18 . . . 4
132, 12bnj832 33250 . . 3
141, 13bnj832 33250 . 2
15 elsni 4058 . . . 4
161, 15bnj833 33251 . . 3
1716fveq2d 5876 . 2
18 bnj602 33408 . . . . . . . 8
1918reseq2d 5279 . . . . . . 7
2016, 19syl 16 . . . . . 6
219bnj931 33264 . . . . . . . . . 10
2221a1i 11 . . . . . . . . 9
23 bnj1442.7 . . . . . . . . . . . 12
24 bnj1442.6 . . . . . . . . . . . . 13
2524simplbi 460 . . . . . . . . . . . 12
2623, 25bnj835 33252 . . . . . . . . . . 11
27 bnj1442.5 . . . . . . . . . . . 12
2827, 23bnj1212 33293 . . . . . . . . . . 11
29 bnj906 33423 . . . . . . . . . . 11
3026, 28, 29syl2anc 661 . . . . . . . . . 10
31 bnj1442.15 . . . . . . . . . . 11
32 fndm 5686 . . . . . . . . . . 11
3331, 32syl 16 . . . . . . . . . 10
3430, 33sseqtr4d 3546 . . . . . . . . 9
354, 22, 34bnj1503 33342 . . . . . . . 8
362, 35bnj832 33250 . . . . . . 7
371, 36bnj832 33250 . . . . . 6
3820, 37eqtrd 2508 . . . . 5
3916, 38opeq12d 4227 . . . 4
40 bnj1442.13 . . . 4
41 bnj1442.11 . . . 4
4239, 40, 413eqtr4g 2533 . . 3
4342fveq2d 5876 . 2
4414, 17, 433eqtr4d 2518 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369   w3a 973   wceq 1379  wex 1596   wcel 1767  cab 2452   wne 2662  wral 2817  wrex 2818  crab 2821  wsbc 3336   cun 3479   wss 3481  c0 3790  csn 4033  cop 4039  cuni 4251   class class class wbr 4453   cdm 5005   cres 5007   wfun 5588   wfn 5589  cfv 5594   c-bnj14 33176   w-bnj15 33180   c-bnj18 33182 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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-reg 8030  ax-inf2 8070 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-reu 2824  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-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  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-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-bnj17 33175  df-bnj14 33177  df-bnj13 33179  df-bnj15 33181  df-bnj18 33183 This theorem is referenced by:  bnj1423  33542
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