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Theorem isoini2 5561
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 5549 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
2 f1of1 5217 . . . . . 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 3209 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐴
75, 6eqsstri 3045 . . . 4 𝐶𝐴
8 f1ores 5233 . . . 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 5560 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)})))
115imaeq2i 4741 . . . . 5 (𝐻𝐶) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋})))
12 isoini2.2 . . . . 5 𝐷 = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)}))
1310, 11, 123eqtr4g 2142 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) = 𝐷)
14 f1oeq3 5211 . . . 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 4992 . . . . . . 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 3074 . . . . . 6 (𝐶𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2120ralimdv 2438 . . . . 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 3074 . . . 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 5294 . . . . . . 7 (𝑥𝐶 → ((𝐻𝐶)‘𝑥) = (𝐻𝑥))
26 fvres 5294 . . . . . . 7 (𝑦𝐶 → ((𝐻𝐶)‘𝑦) = (𝐻𝑦))
2725, 26breqan12d 3837 . . . . . 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 2372 . . . 4 (𝑥𝐶 → (∀𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
3029ralbiia 2388 . . 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 4992 . 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 1287  wcel 1436  wral 2355  cin 2987  wss 2988  {csn 3431   class class class wbr 3822  ccnv 4412  cres 4415  cima 4416  1-1wf1 4980  1-1-ontowf1o 4982  cfv 4983   Isom wiso 4984
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-isom 4992
This theorem is referenced by: (None)
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