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Theorem isoini 7147
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 5931 . 2 (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦}
2 elin 3882 . . . 4 (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})))
3 isof1o 7132 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1ofo 6668 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴onto𝐵)
5 forn 6636 . . . . . . . . . 10 (𝐻:𝐴onto𝐵 → ran 𝐻 = 𝐵)
65eleq2d 2823 . . . . . . . . 9 (𝐻:𝐴onto𝐵 → (𝑦 ∈ ran 𝐻𝑦𝐵))
73, 4, 63syl 18 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦 ∈ ran 𝐻𝑦𝐵))
8 f1ofn 6662 . . . . . . . . 9 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
9 fvelrnb 6773 . . . . . . . . 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 284 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑦𝐵 ↔ ∃𝑥𝐴 (𝐻𝑥) = 𝑦))
12 fvex 6730 . . . . . . . 8 (𝐻𝐷) ∈ V
13 vex 3412 . . . . . . . . 9 𝑦 ∈ V
1413eliniseg 5962 . . . . . . . 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 634 . . . . . 6 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
1716adantr 484 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
18 elin 3882 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})))
19 vex 3412 . . . . . . . . . . . . . 14 𝑥 ∈ V
2019eliniseg 5962 . . . . . . . . . . . . 13 (𝐷𝐴 → (𝑥 ∈ (𝑅 “ {𝐷}) ↔ 𝑥𝑅𝐷))
2120anbi2d 632 . . . . . . . . . . . 12 (𝐷𝐴 → ((𝑥𝐴𝑥 ∈ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2218, 21syl5bb 286 . . . . . . . . . . 11 (𝐷𝐴 → (𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ↔ (𝑥𝐴𝑥𝑅𝐷)))
2322anbi1d 633 . . . . . . . . . 10 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ ((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦)))
24 anass 472 . . . . . . . . . 10 (((𝑥𝐴𝑥𝑅𝐷) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)))
2523, 24bitrdi 290 . . . . . . . . 9 (𝐷𝐴 → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
2625adantl 485 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦))))
27 isorel 7135 . . . . . . . . . . . . . 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 6765 . . . . . . . . . . . . . . . . 17 ((𝐻 Fn 𝐴𝑥𝐴) → ((𝐻𝑥) = 𝑦𝑥𝐻𝑦))
3029bicomd 226 . . . . . . . . . . . . . . . 16 ((𝐻 Fn 𝐴𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3128, 30sylan 583 . . . . . . . . . . . . . . 15 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑥𝐴) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3231adantrr 717 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → (𝑥𝐻𝑦 ↔ (𝐻𝑥) = 𝑦))
3327, 32anbi12d 634 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦)))
34 ancom 464 . . . . . . . . . . . . . 14 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)))
35 breq1 5056 . . . . . . . . . . . . . . 15 ((𝐻𝑥) = 𝑦 → ((𝐻𝑥)𝑆(𝐻𝐷) ↔ 𝑦𝑆(𝐻𝐷)))
3635pm5.32i 578 . . . . . . . . . . . . . 14 (((𝐻𝑥) = 𝑦 ∧ (𝐻𝑥)𝑆(𝐻𝐷)) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3734, 36bitri 278 . . . . . . . . . . . . 13 (((𝐻𝑥)𝑆(𝐻𝐷) ∧ (𝐻𝑥) = 𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
3833, 37bitrdi 290 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝐷𝐴)) → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
3938exp32 424 . . . . . . . . . . 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 410 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑥𝐴 → ((𝑥𝑅𝐷𝑥𝐻𝑦) ↔ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4241pm5.32d 580 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥𝐴 ∧ (𝑥𝑅𝐷𝑥𝐻𝑦)) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4326, 42bitrd 282 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷})) ∧ 𝑥𝐻𝑦) ↔ (𝑥𝐴 ∧ ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))))
4443rexbidv2 3214 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ ∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
45 r19.41v 3260 . . . . . 6 (∃𝑥𝐴 ((𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)) ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷)))
4644, 45bitrdi 290 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦 ↔ (∃𝑥𝐴 (𝐻𝑥) = 𝑦𝑦𝑆(𝐻𝐷))))
4717, 46bitr4d 285 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → ((𝑦𝐵𝑦 ∈ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
482, 47syl5bb 286 . . 3 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝑦 ∈ (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) ↔ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦))
4948abbi2dv 2874 . 2 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ (𝑅 “ {𝐷}))𝑥𝐻𝑦})
501, 49eqtr4id 2797 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 209  wa 399   = wceq 1543  wcel 2110  {cab 2714  wrex 3062  Vcvv 3408  cin 3865  {csn 4541   class class class wbr 5053  ccnv 5550  ran crn 5552  cima 5554   Fn wfn 6375  ontowfo 6378  1-1-ontowf1o 6379  cfv 6380   Isom wiso 6381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389
This theorem is referenced by:  isoini2  7148  isoselem  7150  infxpenlem  9627
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