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Theorem isores3 7281
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 6784 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
2 f1ores 6799 . . . . . . . 8 ((𝐻:𝐴1-1𝐵𝐾𝐴) → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾))
32expcom 415 . . . . . . 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 4011 . . . . . . 7 (𝐾𝐴 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
6 ssralv 4011 . . . . . . . . . 10 (𝐾𝐴 → (∀𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
76adantr 482 . . . . . . . . 9 ((𝐾𝐴𝑎𝐾) → (∀𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
8 fvres 6862 . . . . . . . . . . . . . 14 (𝑎𝐾 → ((𝐻𝐾)‘𝑎) = (𝐻𝑎))
9 fvres 6862 . . . . . . . . . . . . . 14 (𝑏𝐾 → ((𝐻𝐾)‘𝑏) = (𝐻𝑏))
108, 9breqan12d 5122 . . . . . . . . . . . . 13 ((𝑎𝐾𝑏𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1110adantll 713 . . . . . . . . . . . 12 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1211bibi2d 343 . . . . . . . . . . 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 3161 . . . . . . . . 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 3161 . . . . . . 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 610 . . . . 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 6506 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
20 df-isom 6506 . . . . 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 409 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾)))
23 isoeq5 7267 . . 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 1118 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wss 3911   class class class wbr 5106  cres 5636  cima 5637  1-1wf1 6494  1-1-ontowf1o 6496  cfv 6497   Isom wiso 6498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506
This theorem is referenced by:  cantnfp1lem3  9621  fpwwe2lem8  10579  efcvx  25824
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