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Theorem isoini2 5489
Description: Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
Hypotheses
Ref Expression
isoini2.1 𝐶 = (𝐴 ∩ (𝑅 “ {𝑋}))
isoini2.2 𝐷 = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)}))
Assertion
Ref Expression
isoini2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))

Proof of Theorem isoini2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5478 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
2 f1of1 5156 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
31, 2syl 14 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1𝐵)
43adantr 270 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → 𝐻:𝐴1-1𝐵)
5 isoini2.1 . . . . 5 𝐶 = (𝐴 ∩ (𝑅 “ {𝑋}))
6 inss1 3193 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐴
75, 6eqsstri 3030 . . . 4 𝐶𝐴
8 f1ores 5172 . . . 4 ((𝐻:𝐴1-1𝐵𝐶𝐴) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
94, 7, 8sylancl 404 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
10 isoini 5488 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)})))
115imaeq2i 4696 . . . . 5 (𝐻𝐶) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋})))
12 isoini2.2 . . . . 5 𝐷 = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)}))
1310, 11, 123eqtr4g 2139 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) = 𝐷)
14 f1oeq3 5150 . . . 4 ((𝐻𝐶) = 𝐷 → ((𝐻𝐶):𝐶1-1-onto→(𝐻𝐶) ↔ (𝐻𝐶):𝐶1-1-onto𝐷))
1513, 14syl 14 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ((𝐻𝐶):𝐶1-1-onto→(𝐻𝐶) ↔ (𝐻𝐶):𝐶1-1-onto𝐷))
169, 15mpbid 145 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶):𝐶1-1-onto𝐷)
17 df-isom 4941 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
1817simprbi 269 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
1918adantr 270 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
20 ssralv 3059 . . . . . 6 (𝐶𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2120ralimdv 2431 . . . . 5 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑥𝐴𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
227, 19, 21mpsyl 64 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐴𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
23 ssralv 3059 . . . 4 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
247, 22, 23mpsyl 64 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
25 fvres 5230 . . . . . . 7 (𝑥𝐶 → ((𝐻𝐶)‘𝑥) = (𝐻𝑥))
26 fvres 5230 . . . . . . 7 (𝑦𝐶 → ((𝐻𝐶)‘𝑦) = (𝐻𝑦))
2725, 26breqan12d 3808 . . . . . 6 ((𝑥𝐶𝑦𝐶) → (((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦) ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
2827bibi2d 230 . . . . 5 ((𝑥𝐶𝑦𝐶) → ((𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2928ralbidva 2365 . . . 4 (𝑥𝐶 → (∀𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
3029ralbiia 2381 . . 3 (∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
3124, 30sylibr 132 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)))
32 df-isom 4941 . 2 ((𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷) ↔ ((𝐻𝐶):𝐶1-1-onto𝐷 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦))))
3316, 31, 32sylanbrc 408 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wral 2349  cin 2973  wss 2974  {csn 3406   class class class wbr 3793  ccnv 4370  cres 4373  cima 4374  1-1wf1 4929  1-1-ontowf1o 4931  cfv 4932   Isom wiso 4933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-isom 4941
This theorem is referenced by: (None)
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