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Theorem isores3 7355
Description: Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Assertion
Ref Expression
isores3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴𝑋 = (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 6848 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
2 f1ores 6863 . . . . . . . 8 ((𝐻:𝐴1-1𝐵𝐾𝐴) → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾))
32expcom 413 . . . . . . 7 (𝐾𝐴 → (𝐻:𝐴1-1𝐵 → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾)))
41, 3syl5 34 . . . . . 6 (𝐾𝐴 → (𝐻:𝐴1-1-onto𝐵 → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾)))
5 ssralv 4064 . . . . . . 7 (𝐾𝐴 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
6 ssralv 4064 . . . . . . . . . 10 (𝐾𝐴 → (∀𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
76adantr 480 . . . . . . . . 9 ((𝐾𝐴𝑎𝐾) → (∀𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
8 fvres 6926 . . . . . . . . . . . . . 14 (𝑎𝐾 → ((𝐻𝐾)‘𝑎) = (𝐻𝑎))
9 fvres 6926 . . . . . . . . . . . . . 14 (𝑏𝐾 → ((𝐻𝐾)‘𝑏) = (𝐻𝑏))
108, 9breqan12d 5164 . . . . . . . . . . . . 13 ((𝑎𝐾𝑏𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1110adantll 714 . . . . . . . . . . . 12 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1211bibi2d 342 . . . . . . . . . . 11 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → ((𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏)) ↔ (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
1312biimprd 248 . . . . . . . . . 10 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → ((𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
1413ralimdva 3165 . . . . . . . . 9 ((𝐾𝐴𝑎𝐾) → (∀𝑏𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
157, 14syld 47 . . . . . . . 8 ((𝐾𝐴𝑎𝐾) → (∀𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
1615ralimdva 3165 . . . . . . 7 (𝐾𝐴 → (∀𝑎𝐾𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
175, 16syld 47 . . . . . 6 (𝐾𝐴 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
184, 17anim12d 609 . . . . 5 (𝐾𝐴 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))) → ((𝐻𝐾):𝐾1-1-onto→(𝐻𝐾) ∧ ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏)))))
19 df-isom 6572 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
20 df-isom 6572 . . . . 5 ((𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾)) ↔ ((𝐻𝐾):𝐾1-1-onto→(𝐻𝐾) ∧ ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
2118, 19, 203imtr4g 296 . . . 4 (𝐾𝐴 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾))))
2221impcom 407 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾)))
23 isoeq5 7341 . . 3 (𝑋 = (𝐻𝐾) → ((𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋) ↔ (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾))))
2422, 23syl5ibrcom 247 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴) → (𝑋 = (𝐻𝐾) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋)))
25243impia 1116 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴𝑋 = (𝐻𝐾)) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963   class class class wbr 5148  cres 5691  cima 5692  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563   Isom wiso 6564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572
This theorem is referenced by:  cantnfp1lem3  9718  fpwwe2lem8  10676  efcvx  26508
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