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Theorem isoini 5613
Description: Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
Assertion
Ref Expression
isoini ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})))

Proof of Theorem isoini
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3186 . . . 4 (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
2 isof1o 5602 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
3 f1ofo 5275 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
4 forn 5251 . . . . . . . . . 10 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
54eleq2d 2158 . . . . . . . . 9 (𝐻:𝐴onto𝐵 → (𝑦 ∈ ran 𝐻𝑦𝐵))
62, 3, 53syl 17 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻𝑦𝐵))
7 f1ofn 5269 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
8 fvelrnb 5367 . . . . . . . . 9 (𝐻 Fn 𝐴 → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
92, 7, 83syl 17 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
106, 9bitr3d 189 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦𝐵 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
1110adantr 271 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑦𝐵 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
122, 7syl 14 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
1312anim1i 334 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐻 Fn 𝐴𝐷𝐴))
14 funfvex 5337 . . . . . . . 8 ((Fun 𝐻𝐷 ∈ dom 𝐻) → (𝐻𝐷) ∈ V)
1514funfni 5129 . . . . . . 7 ((𝐻 Fn 𝐴𝐷𝐴) → (𝐻𝐷) ∈ V)
16 vex 2625 . . . . . . . 8 𝑦 ∈ V
1716eliniseg 4817 . . . . . . 7 ((𝐻𝐷) ∈ V → (𝑦 ∈ (𝑆 “ {(𝐻𝐷)}) ↔ 𝑦𝑆(𝐻𝐷)))
1813, 15, 173syl 17 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑦 ∈ (𝑆 “ {(𝐻𝐷)}) ↔ 𝑦𝑆(𝐻𝐷)))
1911, 18anbi12d 458 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
20 elin 3186 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})))
21 vex 2625 . . . . . . . . . . . . . 14 𝑥 ∈ V
2221eliniseg 4817 . . . . . . . . . . . . 13 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2322anbi2d 453 . . . . . . . . . . . 12 (𝐷𝐴 → ((𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2420, 23syl5bb 191 . . . . . . . . . . 11 (𝐷𝐴 → (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2524anbi1d 454 . . . . . . . . . 10 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ ((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦)))
26 anass 394 . . . . . . . . . 10 (((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)))
2725, 26syl6bb 195 . . . . . . . . 9 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
2827adantl 272 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
29 isorel 5603 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
30 fnbrfvb 5360 . . . . . . . . . . . . . . . . 17 ((𝐻 Fn 𝐴𝑥𝐴) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
3130bicomd 140 . . . . . . . . . . . . . . . 16 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3212, 31sylan 278 . . . . . . . . . . . . . . 15 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3332adantrr 464 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3429, 33anbi12d 458 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦)))
35 ancom 263 . . . . . . . . . . . . . 14 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)))
36 breq1 3856 . . . . . . . . . . . . . . 15 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
3736pm5.32i 443 . . . . . . . . . . . . . 14 (((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3835, 37bitri 183 . . . . . . . . . . . . 13 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3934, 38syl6bb 195 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
4039exp32 358 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))))
4140com23 78 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐷𝐴 → (𝑥𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))))
4241imp 123 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑥𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4342pm5.32d 439 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4428, 43bitrd 187 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4544rexbidv2 2384 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ ∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
46 r19.41v 2526 . . . . . 6 (∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
4745, 46syl6bb 195 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
4819, 47bitr4d 190 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
491, 48syl5bb 191 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
5049abbi2dv 2207 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦})
51 dfima2 4791 . 2 (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦}
5250, 51syl6reqr 2140 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1290  wcel 1439  {cab 2075  wrex 2361  Vcvv 2622  cin 3001  {csn 3452   class class class wbr 3853  ccnv 4453  ran crn 4455  cima 4457   Fn wfn 5025  ontowfo 5028  1-1-ontowf1o 5029  cfv 5030   Isom wiso 5031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-isom 5039
This theorem is referenced by:  isoini2  5614  isoselem  5615
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