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Theorem isosolem 5384
Description: Lemma for isoso 5385. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
isosolem (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑆 Or B𝑅 Or A))

Proof of Theorem isosolem
Dummy variables 𝑎 𝑏 𝑐 x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isopolem 5382 . . 3 (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑆 Po B𝑅 Po A))
2 df-3an 873 . . . . . . 7 ((𝑎 A 𝑏 A 𝑐 A) ↔ ((𝑎 A 𝑏 A) 𝑐 A))
3 isof1o 5368 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A1-1-ontoB)
4 f1of 5047 . . . . . . . . . . 11 (𝐻:A1-1-ontoB𝐻:AB)
5 ffvelrn 5221 . . . . . . . . . . . . 13 ((𝐻:AB 𝑎 A) → (𝐻𝑎) B)
65ex 108 . . . . . . . . . . . 12 (𝐻:AB → (𝑎 A → (𝐻𝑎) B))
7 ffvelrn 5221 . . . . . . . . . . . . 13 ((𝐻:AB 𝑏 A) → (𝐻𝑏) B)
87ex 108 . . . . . . . . . . . 12 (𝐻:AB → (𝑏 A → (𝐻𝑏) B))
9 ffvelrn 5221 . . . . . . . . . . . . 13 ((𝐻:AB 𝑐 A) → (𝐻𝑐) B)
109ex 108 . . . . . . . . . . . 12 (𝐻:AB → (𝑐 A → (𝐻𝑐) B))
116, 8, 103anim123d 1197 . . . . . . . . . . 11 (𝐻:AB → ((𝑎 A 𝑏 A 𝑐 A) → ((𝐻𝑎) B (𝐻𝑏) B (𝐻𝑐) B)))
123, 4, 113syl 17 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (A, B) → ((𝑎 A 𝑏 A 𝑐 A) → ((𝐻𝑎) B (𝐻𝑏) B (𝐻𝑐) B)))
1312imp 115 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → ((𝐻𝑎) B (𝐻𝑏) B (𝐻𝑐) B))
14 breq1 3737 . . . . . . . . . . 11 (x = (𝐻𝑎) → (x𝑆y ↔ (𝐻𝑎)𝑆y))
15 breq1 3737 . . . . . . . . . . . 12 (x = (𝐻𝑎) → (x𝑆z ↔ (𝐻𝑎)𝑆z))
1615orbi1d 692 . . . . . . . . . . 11 (x = (𝐻𝑎) → ((x𝑆z z𝑆y) ↔ ((𝐻𝑎)𝑆z z𝑆y)))
1714, 16imbi12d 223 . . . . . . . . . 10 (x = (𝐻𝑎) → ((x𝑆y → (x𝑆z z𝑆y)) ↔ ((𝐻𝑎)𝑆y → ((𝐻𝑎)𝑆z z𝑆y))))
18 breq2 3738 . . . . . . . . . . 11 (y = (𝐻𝑏) → ((𝐻𝑎)𝑆y ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
19 breq2 3738 . . . . . . . . . . . 12 (y = (𝐻𝑏) → (z𝑆yz𝑆(𝐻𝑏)))
2019orbi2d 691 . . . . . . . . . . 11 (y = (𝐻𝑏) → (((𝐻𝑎)𝑆z z𝑆y) ↔ ((𝐻𝑎)𝑆z z𝑆(𝐻𝑏))))
2118, 20imbi12d 223 . . . . . . . . . 10 (y = (𝐻𝑏) → (((𝐻𝑎)𝑆y → ((𝐻𝑎)𝑆z z𝑆y)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆z z𝑆(𝐻𝑏)))))
22 breq2 3738 . . . . . . . . . . . 12 (z = (𝐻𝑐) → ((𝐻𝑎)𝑆z ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
23 breq1 3737 . . . . . . . . . . . 12 (z = (𝐻𝑐) → (z𝑆(𝐻𝑏) ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
2422, 23orbi12d 694 . . . . . . . . . . 11 (z = (𝐻𝑐) → (((𝐻𝑎)𝑆z z𝑆(𝐻𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏))))
2524imbi2d 219 . . . . . . . . . 10 (z = (𝐻𝑐) → (((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆z z𝑆(𝐻𝑏))) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏)))))
2617, 21, 25rspc3v 2638 . . . . . . . . 9 (((𝐻𝑎) B (𝐻𝑏) B (𝐻𝑐) B) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏)))))
2713, 26syl 14 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏)))))
28 isorel 5369 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
29283adantr3 1051 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
30 isorel 5369 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑐 A)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
31303adantr2 1050 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
32 isorel 5369 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑐 A 𝑏 A)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3332ancom2s 488 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑏 A 𝑐 A)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
34333adantr1 1049 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3531, 34orbi12d 694 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → ((𝑎𝑅𝑐 𝑐𝑅𝑏) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏))))
3629, 35imbi12d 223 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → ((𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) (𝐻𝑐)𝑆(𝐻𝑏)))))
3727, 36sylibrd 158 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A 𝑐 A)) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
382, 37sylan2br 272 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (A, B) ((𝑎 A 𝑏 A) 𝑐 A)) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
3938anassrs 382 . . . . 5 (((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A)) 𝑐 A) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
4039ralrimdva 2373 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝑎 A 𝑏 A)) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
4140ralrimdvva 2378 . . 3 (𝐻 Isom 𝑅, 𝑆 (A, B) → (x B y B z B (x𝑆y → (x𝑆z z𝑆y)) → 𝑎 A 𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
421, 41anim12d 318 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) → ((𝑆 Po B x B y B z B (x𝑆y → (x𝑆z z𝑆y))) → (𝑅 Po A 𝑎 A 𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏)))))
43 df-iso 4004 . 2 (𝑆 Or B ↔ (𝑆 Po B x B y B z B (x𝑆y → (x𝑆z z𝑆y))))
44 df-iso 4004 . 2 (𝑅 Or A ↔ (𝑅 Po A 𝑎 A 𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
4542, 43, 443imtr4g 194 1 (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑆 Or B𝑅 Or A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 616   w3a 871   = wceq 1226   wcel 1370  wral 2280   class class class wbr 3734   Po wpo 4001   Or wor 4002  wf 4821  1-1-ontowf1o 4824  cfv 4825   Isom wiso 4826
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-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-po 4003  df-iso 4004  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-f1o 4832  df-fv 4833  df-isom 4834
This theorem is referenced by:  isoso  5385
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