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Theorem isoini2 6732
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 6716 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
2 f1of1 6277 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
31, 2syl 17 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1𝐵)
43adantr 466 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → 𝐻:𝐴1-1𝐵)
5 isoini2.1 . . . . 5 𝐶 = (𝐴 ∩ (𝑅 “ {𝑋}))
6 inss1 3981 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐴
75, 6eqsstri 3784 . . . 4 𝐶𝐴
8 f1ores 6292 . . . 4 ((𝐻:𝐴1-1𝐵𝐶𝐴) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
94, 7, 8sylancl 574 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
10 isoini 6731 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)})))
115imaeq2i 5605 . . . . 5 (𝐻𝐶) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋})))
12 isoini2.2 . . . . 5 𝐷 = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)}))
1310, 11, 123eqtr4g 2830 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) = 𝐷)
14 f1oeq3 6270 . . . 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 6040 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
1817simprbi 484 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
1918adantr 466 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
20 ssralv 3815 . . . . . 6 (𝐶𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2120ralimdv 3112 . . . . 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 3815 . . . 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 6348 . . . . . . 7 (𝑥𝐶 → ((𝐻𝐶)‘𝑥) = (𝐻𝑥))
26 fvres 6348 . . . . . . 7 (𝑦𝐶 → ((𝐻𝐶)‘𝑦) = (𝐻𝑦))
2725, 26breqan12d 4802 . . . . . 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 3134 . . . 4 (𝑥𝐶 → (∀𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
3029ralbiia 3128 . . 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 6040 . 2 ((𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷) ↔ ((𝐻𝐶):𝐶1-1-onto𝐷 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦))))
3316, 31, 32sylanbrc 572 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 382   = wceq 1631  wcel 2145  wral 3061  cin 3722  wss 3723  {csn 4316   class class class wbr 4786  ccnv 5248  cres 5251  cima 5252  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031   Isom wiso 6032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040
This theorem is referenced by:  fz1isolem  13447
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