MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isoini2 Structured version   Visualization version   GIF version

Theorem isoini2 6629
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 6613 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
2 f1of1 6174 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
31, 2syl 17 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1𝐵)
43adantr 480 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → 𝐻:𝐴1-1𝐵)
5 isoini2.1 . . . . 5 𝐶 = (𝐴 ∩ (𝑅 “ {𝑋}))
6 inss1 3866 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐴
75, 6eqsstri 3668 . . . 4 𝐶𝐴
8 f1ores 6189 . . . 4 ((𝐻:𝐴1-1𝐵𝐶𝐴) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
94, 7, 8sylancl 695 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
10 isoini 6628 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)})))
115imaeq2i 5499 . . . . 5 (𝐻𝐶) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋})))
12 isoini2.2 . . . . 5 𝐷 = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)}))
1310, 11, 123eqtr4g 2710 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) = 𝐷)
14 f1oeq3 6167 . . . 4 ((𝐻𝐶) = 𝐷 → ((𝐻𝐶):𝐶1-1-onto→(𝐻𝐶) ↔ (𝐻𝐶):𝐶1-1-onto𝐷))
1513, 14syl 17 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ((𝐻𝐶):𝐶1-1-onto→(𝐻𝐶) ↔ (𝐻𝐶):𝐶1-1-onto𝐷))
169, 15mpbid 222 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶):𝐶1-1-onto𝐷)
17 df-isom 5935 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
1817simprbi 479 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
1918adantr 480 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
20 ssralv 3699 . . . . . 6 (𝐶𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2120ralimdv 2992 . . . . 5 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑥𝐴𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
227, 19, 21mpsyl 68 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐴𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
23 ssralv 3699 . . . 4 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
247, 22, 23mpsyl 68 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
25 fvres 6245 . . . . . . 7 (𝑥𝐶 → ((𝐻𝐶)‘𝑥) = (𝐻𝑥))
26 fvres 6245 . . . . . . 7 (𝑦𝐶 → ((𝐻𝐶)‘𝑦) = (𝐻𝑦))
2725, 26breqan12d 4701 . . . . . 6 ((𝑥𝐶𝑦𝐶) → (((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦) ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
2827bibi2d 331 . . . . 5 ((𝑥𝐶𝑦𝐶) → ((𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2928ralbidva 3014 . . . 4 (𝑥𝐶 → (∀𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
3029ralbiia 3008 . . 3 (∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
3124, 30sylibr 224 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)))
32 df-isom 5935 . 2 ((𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷) ↔ ((𝐻𝐶):𝐶1-1-onto𝐷 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦))))
3316, 31, 32sylanbrc 699 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  cin 3606  wss 3607  {csn 4210   class class class wbr 4685  ccnv 5142  cres 5145  cima 5146  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935
This theorem is referenced by:  fz1isolem  13283
  Copyright terms: Public domain W3C validator