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Theorem isores3 5368
 Description: Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Assertion
Ref Expression
isores3 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐾A 𝑋 = (𝐻𝐾)) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))

Proof of Theorem isores3
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1of1 5038 . . . . . . 7 (𝐻:A1-1-ontoB𝐻:A1-1B)
2 f1ores 5054 . . . . . . . 8 ((𝐻:A1-1B 𝐾A) → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾))
32expcom 109 . . . . . . 7 (𝐾A → (𝐻:A1-1B → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾)))
41, 3syl5 28 . . . . . 6 (𝐾A → (𝐻:A1-1-ontoB → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾)))
5 ssralv 2972 . . . . . . 7 (𝐾A → (𝑎 A 𝑏 A (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → 𝑎 𝐾 𝑏 A (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
6 ssralv 2972 . . . . . . . . . 10 (𝐾A → (𝑏 A (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → 𝑏 𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
76adantr 261 . . . . . . . . 9 ((𝐾A 𝑎 𝐾) → (𝑏 A (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → 𝑏 𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
8 fvres 5111 . . . . . . . . . . . . . 14 (𝑎 𝐾 → ((𝐻𝐾)‘𝑎) = (𝐻𝑎))
9 fvres 5111 . . . . . . . . . . . . . 14 (𝑏 𝐾 → ((𝐻𝐾)‘𝑏) = (𝐻𝑏))
108, 9breqan12d 3742 . . . . . . . . . . . . 13 ((𝑎 𝐾 𝑏 𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1110adantll 445 . . . . . . . . . . . 12 (((𝐾A 𝑎 𝐾) 𝑏 𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1211bibi2d 221 . . . . . . . . . . 11 (((𝐾A 𝑎 𝐾) 𝑏 𝐾) → ((𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏)) ↔ (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
1312biimprd 147 . . . . . . . . . 10 (((𝐾A 𝑎 𝐾) 𝑏 𝐾) → ((𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
1413ralimdva 2356 . . . . . . . . 9 ((𝐾A 𝑎 𝐾) → (𝑏 𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → 𝑏 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
157, 14syld 40 . . . . . . . 8 ((𝐾A 𝑎 𝐾) → (𝑏 A (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → 𝑏 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
1615ralimdva 2356 . . . . . . 7 (𝐾A → (𝑎 𝐾 𝑏 A (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → 𝑎 𝐾 𝑏 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
175, 16syld 40 . . . . . 6 (𝐾A → (𝑎 A 𝑏 A (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → 𝑎 𝐾 𝑏 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
184, 17anim12d 318 . . . . 5 (𝐾A → ((𝐻:A1-1-ontoB 𝑎 A 𝑏 A (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))) → ((𝐻𝐾):𝐾1-1-onto→(𝐻𝐾) 𝑎 𝐾 𝑏 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏)))))
19 df-isom 4826 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB 𝑎 A 𝑏 A (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
20 df-isom 4826 . . . . 5 ((𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾)) ↔ ((𝐻𝐾):𝐾1-1-onto→(𝐻𝐾) 𝑎 𝐾 𝑏 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
2118, 19, 203imtr4g 194 . . . 4 (𝐾A → (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾))))
2221impcom 116 . . 3 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐾A) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾)))
23 isoeq5 5358 . . 3 (𝑋 = (𝐻𝐾) → ((𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋) ↔ (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾))))
2422, 23syl5ibrcom 146 . 2 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐾A) → (𝑋 = (𝐻𝐾) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋)))
25243impia 1082 1 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐾A 𝑋 = (𝐻𝐾)) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 867   = wceq 1223   ∈ wcel 1366  ∀wral 2275   ⊆ wss 2885   class class class wbr 3727   ↾ cres 4262   “ cima 4263  –1-1→wf1 4814  –1-1-onto→wf1o 4816  ‘cfv 4817   Isom wiso 4818 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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-opab 3782  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-isom 4826 This theorem is referenced by: (None)
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