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Theorem isoini2 5401
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) 𝑋 A) → (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 5390 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A1-1-ontoB)
2 f1of1 5068 . . . . . 6 (𝐻:A1-1-ontoB𝐻:A1-1B)
31, 2syl 14 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A1-1B)
43adantr 261 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → 𝐻:A1-1B)
5 isoini2.1 . . . . 5 𝐶 = (A ∩ (𝑅 “ {𝑋}))
6 inss1 3151 . . . . 5 (A ∩ (𝑅 “ {𝑋})) ⊆ A
75, 6eqsstri 2969 . . . 4 𝐶A
8 f1ores 5084 . . . 4 ((𝐻:A1-1B 𝐶A) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
94, 7, 8sylancl 392 . . 3 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
10 isoini 5400 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → (𝐻 “ (A ∩ (𝑅 “ {𝑋}))) = (B ∩ (𝑆 “ {(𝐻𝑋)})))
115imaeq2i 4609 . . . . 5 (𝐻𝐶) = (𝐻 “ (A ∩ (𝑅 “ {𝑋})))
12 isoini2.2 . . . . 5 𝐷 = (B ∩ (𝑆 “ {(𝐻𝑋)}))
1310, 11, 123eqtr4g 2094 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → (𝐻𝐶) = 𝐷)
14 f1oeq3 5062 . . . 4 ((𝐻𝐶) = 𝐷 → ((𝐻𝐶):𝐶1-1-onto→(𝐻𝐶) ↔ (𝐻𝐶):𝐶1-1-onto𝐷))
1513, 14syl 14 . . 3 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → ((𝐻𝐶):𝐶1-1-onto→(𝐻𝐶) ↔ (𝐻𝐶):𝐶1-1-onto𝐷))
169, 15mpbid 135 . 2 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → (𝐻𝐶):𝐶1-1-onto𝐷)
17 df-isom 4854 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
1817simprbi 260 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (A, B) → x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)))
1918adantr 261 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)))
20 ssralv 2998 . . . . . 6 (𝐶A → (y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) → y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
2120ralimdv 2382 . . . . 5 (𝐶A → (x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) → x A y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
227, 19, 21mpsyl 59 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → x A y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y)))
23 ssralv 2998 . . . 4 (𝐶A → (x A y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) → x 𝐶 y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
247, 22, 23mpsyl 59 . . 3 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → x 𝐶 y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y)))
25 fvres 5141 . . . . . . 7 (x 𝐶 → ((𝐻𝐶)‘x) = (𝐻x))
26 fvres 5141 . . . . . . 7 (y 𝐶 → ((𝐻𝐶)‘y) = (𝐻y))
2725, 26breqan12d 3770 . . . . . 6 ((x 𝐶 y 𝐶) → (((𝐻𝐶)‘x)𝑆((𝐻𝐶)‘y) ↔ (𝐻x)𝑆(𝐻y)))
2827bibi2d 221 . . . . 5 ((x 𝐶 y 𝐶) → ((x𝑅y ↔ ((𝐻𝐶)‘x)𝑆((𝐻𝐶)‘y)) ↔ (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
2928ralbidva 2316 . . . 4 (x 𝐶 → (y 𝐶 (x𝑅y ↔ ((𝐻𝐶)‘x)𝑆((𝐻𝐶)‘y)) ↔ y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
3029ralbiia 2332 . . 3 (x 𝐶 y 𝐶 (x𝑅y ↔ ((𝐻𝐶)‘x)𝑆((𝐻𝐶)‘y)) ↔ x 𝐶 y 𝐶 (x𝑅y ↔ (𝐻x)𝑆(𝐻y)))
3124, 30sylibr 137 . 2 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → x 𝐶 y 𝐶 (x𝑅y ↔ ((𝐻𝐶)‘x)𝑆((𝐻𝐶)‘y)))
32 df-isom 4854 . 2 ((𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷) ↔ ((𝐻𝐶):𝐶1-1-onto𝐷 x 𝐶 y 𝐶 (x𝑅y ↔ ((𝐻𝐶)‘x)𝑆((𝐻𝐶)‘y))))
3316, 31, 32sylanbrc 394 1 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝑋 A) → (𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  cin 2910  wss 2911  {csn 3367   class class class wbr 3755  ccnv 4287  cres 4290  cima 4291  1-1wf1 4842  1-1-ontowf1o 4844  cfv 4845   Isom wiso 4846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-isom 4854
This theorem is referenced by: (None)
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