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Theorem isores3 5632
 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 5287 . . . . . . 7 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
2 f1ores 5303 . . . . . . . 8 ((𝐻:𝐴1-1𝐵𝐾𝐴) → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾))
32expcom 115 . . . . . . 7 (𝐾𝐴 → (𝐻:𝐴1-1𝐵 → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾)))
41, 3syl5 32 . . . . . 6 (𝐾𝐴 → (𝐻:𝐴1-1-onto𝐵 → (𝐻𝐾):𝐾1-1-onto→(𝐻𝐾)))
5 ssralv 3100 . . . . . . 7 (𝐾𝐴 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
6 ssralv 3100 . . . . . . . . . 10 (𝐾𝐴 → (∀𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
76adantr 271 . . . . . . . . 9 ((𝐾𝐴𝑎𝐾) → (∀𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
8 fvres 5364 . . . . . . . . . . . . . 14 (𝑎𝐾 → ((𝐻𝐾)‘𝑎) = (𝐻𝑎))
9 fvres 5364 . . . . . . . . . . . . . 14 (𝑏𝐾 → ((𝐻𝐾)‘𝑏) = (𝐻𝑏))
108, 9breqan12d 3882 . . . . . . . . . . . . 13 ((𝑎𝐾𝑏𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1110adantll 461 . . . . . . . . . . . 12 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → (((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏) ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
1211bibi2d 231 . . . . . . . . . . 11 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → ((𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏)) ↔ (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
1312biimprd 157 . . . . . . . . . 10 (((𝐾𝐴𝑎𝐾) ∧ 𝑏𝐾) → ((𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
1413ralimdva 2453 . . . . . . . . 9 ((𝐾𝐴𝑎𝐾) → (∀𝑏𝐾 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
157, 14syld 45 . . . . . . . 8 ((𝐾𝐴𝑎𝐾) → (∀𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
1615ralimdva 2453 . . . . . . 7 (𝐾𝐴 → (∀𝑎𝐾𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
175, 16syld 45 . . . . . 6 (𝐾𝐴 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)) → ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
184, 17anim12d 329 . . . . 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 5058 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏))))
20 df-isom 5058 . . . . 5 ((𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾)) ↔ ((𝐻𝐾):𝐾1-1-onto→(𝐻𝐾) ∧ ∀𝑎𝐾𝑏𝐾 (𝑎𝑅𝑏 ↔ ((𝐻𝐾)‘𝑎)𝑆((𝐻𝐾)‘𝑏))))
2118, 19, 203imtr4g 204 . . . 4 (𝐾𝐴 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾))))
2221impcom 124 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾)))
23 isoeq5 5622 . . 3 (𝑋 = (𝐻𝐾) → ((𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋) ↔ (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻𝐾))))
2422, 23syl5ibrcom 156 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴) → (𝑋 = (𝐻𝐾) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋)))
25243impia 1143 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴𝑋 = (𝐻𝐾)) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 927   = wceq 1296   ∈ wcel 1445  ∀wral 2370   ⊆ wss 3013   class class class wbr 3867   ↾ cres 4469   “ cima 4470  –1-1→wf1 5046  –1-1-onto→wf1o 5048  ‘cfv 5049   Isom wiso 5050 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-isom 5058 This theorem is referenced by: (None)
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