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Theorem isores3 6485
 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 6049 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
2 f1ores 6064 . . . . . . . 8 ((𝐻:𝐴1-1𝐵𝐾𝐴) → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾))
32expcom 450 . . . . . . 7 (𝐾𝐴 → (𝐻:𝐴1-1𝐵 → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾)))
41, 3syl5 33 . . . . . 6 (𝐾𝐴 → (𝐻:𝐴1-1-onto𝐵 → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾)))
5 ssralv 3629 . . . . . . 7 (𝐾𝐴 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
6 ssralv 3629 . . . . . . . . . 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 6117 . . . . . . . . . . . . . 14 (𝑎𝐾 → ((𝐻𝐾)‘𝑎) = (𝐻𝑎))
9 fvres 6117 . . . . . . . . . . . . . 14 (𝑏𝐾 → ((𝐻𝐾)‘𝑏) = (𝐻𝑏))
108, 9breqan12d 4599 . . . . . . . . . . . . 13 ((𝑎𝐾𝑏𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1110adantll 746 . . . . . . . . . . . 12 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1211bibi2d 331 . . . . . . . . . . 11 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → ((𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏)) ↔ (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
1312biimprd 237 . . . . . . . . . 10 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → ((𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
1413ralimdva 2945 . . . . . . . . 9 ((𝐾𝐴𝑎𝐾) → (∀𝑏𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
157, 14syld 46 . . . . . . . 8 ((𝐾𝐴𝑎𝐾) → (∀𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
1615ralimdva 2945 . . . . . . 7 (𝐾𝐴 → (∀𝑎𝐾𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
175, 16syld 46 . . . . . 6 (𝐾𝐴 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
184, 17anim12d 584 . . . . 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 5813 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
20 df-isom 5813 . . . . 5 ((𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾)) ↔ ((𝐻𝐾):𝐾1-1-onto→(𝐻𝐾) ∧ ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
2118, 19, 203imtr4g 284 . . . 4 (𝐾𝐴 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾))))
2221impcom 445 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾)))
23 isoeq5 6471 . . 3 (𝑋 = (𝐻𝐾) → ((𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋) ↔ (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾))))
2422, 23syl5ibrcom 236 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴) → (𝑋 = (𝐻𝐾) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋)))
25243impia 1253 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   class class class wbr 4583   ↾ cres 5040   “ cima 5041  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813 This theorem is referenced by:  cantnfp1lem3  8460  fpwwe2lem9  9339  efcvx  24007
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