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Theorem isoini2 5798
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 5786 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
2 f1of1 5441 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
31, 2syl 14 . . . . 5 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1𝐵)
43adantr 274 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → 𝐻:𝐴1-1𝐵)
5 isoini2.1 . . . . 5 𝐶 = (𝐴 ∩ (𝑅 “ {𝑋}))
6 inss1 3347 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐴
75, 6eqsstri 3179 . . . 4 𝐶𝐴
8 f1ores 5457 . . . 4 ((𝐻:𝐴1-1𝐵𝐶𝐴) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
94, 7, 8sylancl 411 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶):𝐶1-1-onto→(𝐻𝐶))
10 isoini 5797 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)})))
115imaeq2i 4951 . . . . 5 (𝐻𝐶) = (𝐻 “ (𝐴 ∩ (𝑅 “ {𝑋})))
12 isoini2.2 . . . . 5 𝐷 = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)}))
1310, 11, 123eqtr4g 2228 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) = 𝐷)
14 f1oeq3 5433 . . . 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 146 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶):𝐶1-1-onto𝐷)
17 df-isom 5207 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
1817simprbi 273 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
1918adantr 274 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
20 ssralv 3211 . . . . . 6 (𝐶𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2120ralimdv 2538 . . . . 5 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑥𝐴𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
227, 19, 21mpsyl 65 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐴𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
23 ssralv 3211 . . . 4 (𝐶𝐴 → (∀𝑥𝐴𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
247, 22, 23mpsyl 65 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
25 fvres 5520 . . . . . . 7 (𝑥𝐶 → ((𝐻𝐶)‘𝑥) = (𝐻𝑥))
26 fvres 5520 . . . . . . 7 (𝑦𝐶 → ((𝐻𝐶)‘𝑦) = (𝐻𝑦))
2725, 26breqan12d 4005 . . . . . 6 ((𝑥𝐶𝑦𝐶) → (((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦) ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
2827bibi2d 231 . . . . 5 ((𝑥𝐶𝑦𝐶) → ((𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2928ralbidva 2466 . . . 4 (𝑥𝐶 → (∀𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
3029ralbiia 2484 . . 3 (∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)) ↔ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
3124, 30sylibr 133 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦)))
32 df-isom 5207 . 2 ((𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷) ↔ ((𝐻𝐶):𝐶1-1-onto𝐷 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ ((𝐻𝐶)‘𝑥)𝑆((𝐻𝐶)‘𝑦))))
3316, 31, 32sylanbrc 415 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 103  wb 104   = wceq 1348  wcel 2141  wral 2448  cin 3120  wss 3121  {csn 3583   class class class wbr 3989  ccnv 4610  cres 4613  cima 4614  1-1wf1 5195  1-1-ontowf1o 5197  cfv 5198   Isom wiso 5199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207
This theorem is referenced by: (None)
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