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Theorem isoini 6628
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 3829 . . . 4 (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
2 isof1o 6613 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
3 f1ofo 6182 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
4 forn 6156 . . . . . . . . . 10 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
54eleq2d 2716 . . . . . . . . 9 (𝐻:𝐴onto𝐵 → (𝑦 ∈ ran 𝐻𝑦𝐵))
62, 3, 53syl 18 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻𝑦𝐵))
7 f1ofn 6176 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
8 fvelrnb 6282 . . . . . . . . 9 (𝐻 Fn 𝐴 → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
92, 7, 83syl 18 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
106, 9bitr3d 270 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦𝐵 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
11 fvex 6239 . . . . . . . 8 (𝐻𝐷) ∈ V
12 vex 3234 . . . . . . . . 9 𝑦 ∈ V
1312eliniseg 5529 . . . . . . . 8 ((𝐻𝐷) ∈ V → (𝑦 ∈ (𝑆 “ {(𝐻𝐷)}) ↔ 𝑦𝑆(𝐻𝐷)))
1411, 13mp1i 13 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ (𝑆 “ {(𝐻𝐷)}) ↔ 𝑦𝑆(𝐻𝐷)))
1510, 14anbi12d 747 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
1615adantr 480 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
17 elin 3829 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})))
18 vex 3234 . . . . . . . . . . . . . 14 𝑥 ∈ V
1918eliniseg 5529 . . . . . . . . . . . . 13 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2019anbi2d 740 . . . . . . . . . . . 12 (𝐷𝐴 → ((𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2117, 20syl5bb 272 . . . . . . . . . . 11 (𝐷𝐴 → (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2221anbi1d 741 . . . . . . . . . 10 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ ((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦)))
23 anass 682 . . . . . . . . . 10 (((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)))
2422, 23syl6bb 276 . . . . . . . . 9 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
2524adantl 481 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
26 isorel 6616 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
272, 7syl 17 . . . . . . . . . . . . . . . 16 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
28 fnbrfvb 6274 . . . . . . . . . . . . . . . . 17 ((𝐻 Fn 𝐴𝑥𝐴) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
2928bicomd 213 . . . . . . . . . . . . . . . 16 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3027, 29sylan 487 . . . . . . . . . . . . . . 15 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3130adantrr 753 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3226, 31anbi12d 747 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦)))
33 ancom 465 . . . . . . . . . . . . . 14 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)))
34 breq1 4688 . . . . . . . . . . . . . . 15 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
3534pm5.32i 670 . . . . . . . . . . . . . 14 (((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3633, 35bitri 264 . . . . . . . . . . . . 13 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3732, 36syl6bb 276 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
3837exp32 630 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))))
3938com23 86 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐷𝐴 → (𝑥𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))))
4039imp 444 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑥𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4140pm5.32d 672 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4225, 41bitrd 268 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4342rexbidv2 3077 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ ∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
44 r19.41v 3118 . . . . . 6 (∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
4543, 44syl6bb 276 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
4616, 45bitr4d 271 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
471, 46syl5bb 272 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
4847abbi2dv 2771 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦})
49 dfima2 5503 . 2 (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦}
5048, 49syl6reqr 2704 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  Vcvv 3231  cin 3606  {csn 4210   class class class wbr 4685  ccnv 5142  ran crn 5144  cima 5146   Fn wfn 5921  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935
This theorem is referenced by:  isoini2  6629  isoselem  6631  infxpenlem  8874
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