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Theorem isoini 7335
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 dfima2 6062 . 2 (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦}
2 elin 3965 . . . 4 (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
3 isof1o 7320 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1ofo 6841 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
5 forn 6809 . . . . . . . . . 10 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
65eleq2d 2820 . . . . . . . . 9 (𝐻:𝐴onto𝐵 → (𝑦 ∈ ran 𝐻𝑦𝐵))
73, 4, 63syl 18 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻𝑦𝐵))
8 f1ofn 6835 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
9 fvelrnb 6953 . . . . . . . . 9 (𝐻 Fn 𝐴 → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
103, 8, 93syl 18 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
117, 10bitr3d 281 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦𝐵 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
12 fvex 6905 . . . . . . . 8 (𝐻𝐷) ∈ V
13 vex 3479 . . . . . . . . 9 𝑦 ∈ V
1413eliniseg 6094 . . . . . . . 8 ((𝐻𝐷) ∈ V → (𝑦 ∈ (𝑆 “ {(𝐻𝐷)}) ↔ 𝑦𝑆(𝐻𝐷)))
1512, 14mp1i 13 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ (𝑆 “ {(𝐻𝐷)}) ↔ 𝑦𝑆(𝐻𝐷)))
1611, 15anbi12d 632 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
1716adantr 482 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
18 elin 3965 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})))
19 vex 3479 . . . . . . . . . . . . . 14 𝑥 ∈ V
2019eliniseg 6094 . . . . . . . . . . . . 13 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2120anbi2d 630 . . . . . . . . . . . 12 (𝐷𝐴 → ((𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2218, 21bitrid 283 . . . . . . . . . . 11 (𝐷𝐴 → (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2322anbi1d 631 . . . . . . . . . 10 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ ((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦)))
24 anass 470 . . . . . . . . . 10 (((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)))
2523, 24bitrdi 287 . . . . . . . . 9 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
2625adantl 483 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
27 isorel 7323 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝑅𝐷 ↔ (𝐻𝑥)𝑆(𝐻𝐷)))
283, 8syl 17 . . . . . . . . . . . . . . . 16 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
29 fnbrfvb 6945 . . . . . . . . . . . . . . . . 17 ((𝐻 Fn 𝐴𝑥𝐴) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
3029bicomd 222 . . . . . . . . . . . . . . . 16 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3128, 30sylan 581 . . . . . . . . . . . . . . 15 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3231adantrr 716 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3327, 32anbi12d 632 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦)))
34 ancom 462 . . . . . . . . . . . . . 14 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)))
35 breq1 5152 . . . . . . . . . . . . . . 15 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
3635pm5.32i 576 . . . . . . . . . . . . . 14 (((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3734, 36bitri 275 . . . . . . . . . . . . 13 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3833, 37bitrdi 287 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
3938exp32 422 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑥𝐴 → (𝐷𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))))
4039com23 86 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐷𝐴 → (𝑥𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))))
4140imp 408 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑥𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4241pm5.32d 578 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4326, 42bitrd 279 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4443rexbidv2 3175 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ ∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
45 r19.41v 3189 . . . . . 6 (∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
4644, 45bitrdi 287 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
4717, 46bitr4d 282 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
482, 47bitrid 283 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
4948eqabdv 2868 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦})
501, 49eqtr4id 2792 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 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  Vcvv 3475  cin 3948  {csn 4629   class class class wbr 5149  ccnv 5676  ran crn 5678  cima 5680   Fn wfn 6539  ontowfo 6542  1-1-ontowf1o 6543  cfv 6544   Isom wiso 6545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553
This theorem is referenced by:  isoini2  7336  isoselem  7338  infxpenlem  10008
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