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Theorem isoini 5484
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 3153 . . . 4 (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
2 isof1o 5474 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
3 f1ofo 5160 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
4 forn 5136 . . . . . . . . . 10 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
54eleq2d 2123 . . . . . . . . 9 (𝐻:𝐴onto𝐵 → (𝑦 ∈ ran 𝐻𝑦𝐵))
62, 3, 53syl 17 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻𝑦𝐵))
7 f1ofn 5154 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
8 fvelrnb 5248 . . . . . . . . 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 183 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦𝐵 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
1110adantr 265 . . . . . 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 327 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐻 Fn 𝐴𝐷𝐴))
14 funfvex 5219 . . . . . . . 8 ((Fun 𝐻𝐷 ∈ dom 𝐻) → (𝐻𝐷) ∈ V)
1514funfni 5026 . . . . . . 7 ((𝐻 Fn 𝐴𝐷𝐴) → (𝐻𝐷) ∈ V)
16 vex 2577 . . . . . . . 8 𝑦 ∈ V
1716eliniseg 4722 . . . . . . 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 450 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
20 elin 3153 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})))
21 vex 2577 . . . . . . . . . . . . . 14 𝑥 ∈ V
2221eliniseg 4722 . . . . . . . . . . . . 13 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2322anbi2d 445 . . . . . . . . . . . 12 (𝐷𝐴 → ((𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2420, 23syl5bb 185 . . . . . . . . . . 11 (𝐷𝐴 → (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2524anbi1d 446 . . . . . . . . . 10 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ ((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦)))
26 anass 387 . . . . . . . . . 10 (((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)))
2725, 26syl6bb 189 . . . . . . . . 9 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
2827adantl 266 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
29 isorel 5475 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
30 fnbrfvb 5241 . . . . . . . . . . . . . . . . 17 ((𝐻 Fn 𝐴𝑥𝐴) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
3130bicomd 133 . . . . . . . . . . . . . . . 16 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3212, 31sylan 271 . . . . . . . . . . . . . . 15 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3332adantrr 456 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3429, 33anbi12d 450 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦)))
35 ancom 257 . . . . . . . . . . . . . 14 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)))
36 breq1 3794 . . . . . . . . . . . . . . 15 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
3736pm5.32i 435 . . . . . . . . . . . . . 14 (((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3835, 37bitri 177 . . . . . . . . . . . . 13 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3934, 38syl6bb 189 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
4039exp32 351 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))))
4140com23 76 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐷𝐴 → (𝑥𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))))
4241imp 119 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑥𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4342pm5.32d 431 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4428, 43bitrd 181 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4544rexbidv2 2346 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ ∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
46 r19.41v 2483 . . . . . 6 (∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
4745, 46syl6bb 189 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
4819, 47bitr4d 184 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
491, 48syl5bb 185 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
5049abbi2dv 2172 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦})
51 dfima2 4697 . 2 (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦}
5250, 51syl6reqr 2107 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 101  wb 102   = wceq 1259  wcel 1409  {cab 2042  wrex 2324  Vcvv 2574  cin 2943  {csn 3402   class class class wbr 3791  ccnv 4371  ran crn 4373  cima 4375   Fn wfn 4924  ontowfo 4927  1-1-ontowf1o 4928  cfv 4929   Isom wiso 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-isom 4938
This theorem is referenced by:  isoini2  5485  isoselem  5486
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