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Theorem bnj1442 29480
 Description: Technical lemma for bnj60 29493. 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 29202 . . . . 5
5 opex 4429 . . . . . . . 8
65snid 3843 . . . . . . 7
7 elun2 3517 . . . . . . 7
86, 7ax-mp 8 . . . . . 6
9 bnj1442.12 . . . . . 6
108, 9eleqtrri 2511 . . . . 5
11 funopfv 5768 . . . . 5
124, 10, 11ee10 1386 . . . 4
132, 12bnj832 29188 . . 3
141, 13bnj832 29188 . 2
15 elsni 3840 . . . 4
161, 15bnj833 29189 . . 3
1716fveq2d 5734 . 2
18 bnj602 29348 . . . . . . . 8
1918reseq2d 5148 . . . . . . 7
2016, 19syl 16 . . . . . 6
219bnj931 29203 . . . . . . . . . 10
2221a1i 11 . . . . . . . . 9
23 bnj1442.7 . . . . . . . . . . . 12
24 bnj1442.6 . . . . . . . . . . . . 13
2524simplbi 448 . . . . . . . . . . . 12
2623, 25bnj835 29190 . . . . . . . . . . 11
27 bnj1442.5 . . . . . . . . . . . 12
2827, 23bnj1212 29233 . . . . . . . . . . 11
29 bnj906 29363 . . . . . . . . . . 11
3026, 28, 29syl2anc 644 . . . . . . . . . 10
31 bnj1442.15 . . . . . . . . . . 11
32 fndm 5546 . . . . . . . . . . 11
3331, 32syl 16 . . . . . . . . . 10
3430, 33sseqtr4d 3387 . . . . . . . . 9
354, 22, 34bnj1503 29282 . . . . . . . 8
362, 35bnj832 29188 . . . . . . 7
371, 36bnj832 29188 . . . . . 6
3820, 37eqtrd 2470 . . . . 5
3916, 38opeq12d 3994 . . . 4
40 bnj1442.13 . . . 4
41 bnj1442.11 . . . 4
4239, 40, 413eqtr4g 2495 . . 3
4342fveq2d 5734 . 2
4414, 17, 433eqtr4d 2480 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360   w3a 937  wex 1551   wceq 1653   wcel 1726  cab 2424   wne 2601  wral 2707  wrex 2708  crab 2711  wsbc 3163   cun 3320   wss 3322  c0 3630  csn 3816  cop 3819  cuni 4017   class class class wbr 4214   cdm 4880   cres 4882   wfun 5450   wfn 5451  cfv 5456   c-bnj14 29114   w-bnj15 29118   c-bnj18 29120 This theorem is referenced by:  bnj1423  29482 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-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-reg 7562  ax-inf2 7598 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-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-1o 6726  df-bnj17 29113  df-bnj14 29115  df-bnj13 29117  df-bnj15 29119  df-bnj18 29121
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