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Theorem isoini 5400
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) 𝐷 A) → (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 3120 . . . 4 (y (B ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (y B y (𝑆 “ {(𝐻𝐷)})))
2 isof1o 5390 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A1-1-ontoB)
3 f1ofo 5076 . . . . . . . . 9 (𝐻:A1-1-ontoB𝐻:AontoB)
4 forn 5052 . . . . . . . . . 10 (𝐻:AontoB → ran 𝐻 = B)
54eleq2d 2104 . . . . . . . . 9 (𝐻:AontoB → (y ran 𝐻y B))
62, 3, 53syl 17 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (A, B) → (y ran 𝐻y B))
7 f1ofn 5070 . . . . . . . . 9 (𝐻:A1-1-ontoB𝐻 Fn A)
8 fvelrnb 5164 . . . . . . . . 9 (𝐻 Fn A → (y ran 𝐻x A (𝐻x) = y))
92, 7, 83syl 17 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (A, B) → (y ran 𝐻x A (𝐻x) = y))
106, 9bitr3d 179 . . . . . . 7 (𝐻 Isom 𝑅, 𝑆 (A, B) → (y Bx A (𝐻x) = y))
1110adantr 261 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (y Bx A (𝐻x) = y))
122, 7syl 14 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻 Fn A)
1312anim1i 323 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (𝐻 Fn A 𝐷 A))
14 funfvex 5135 . . . . . . . 8 ((Fun 𝐻 𝐷 dom 𝐻) → (𝐻𝐷) V)
1514funfni 4942 . . . . . . 7 ((𝐻 Fn A 𝐷 A) → (𝐻𝐷) V)
16 vex 2554 . . . . . . . 8 y V
1716eliniseg 4638 . . . . . . 7 ((𝐻𝐷) V → (y (𝑆 “ {(𝐻𝐷)}) ↔ y𝑆(𝐻𝐷)))
1813, 15, 173syl 17 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (y (𝑆 “ {(𝐻𝐷)}) ↔ y𝑆(𝐻𝐷)))
1911, 18anbi12d 442 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → ((y B y (𝑆 “ {(𝐻𝐷)})) ↔ (x A (𝐻x) = y y𝑆(𝐻𝐷))))
20 elin 3120 . . . . . . . . . . . 12 (x (A ∩ (𝑅 “ {𝐷})) ↔ (x A x (𝑅 “ {𝐷})))
21 vex 2554 . . . . . . . . . . . . . 14 x V
2221eliniseg 4638 . . . . . . . . . . . . 13 (𝐷 A → (x (𝑅 “ {𝐷}) ↔ x𝑅𝐷))
2322anbi2d 437 . . . . . . . . . . . 12 (𝐷 A → ((x A x (𝑅 “ {𝐷})) ↔ (x A x𝑅𝐷)))
2420, 23syl5bb 181 . . . . . . . . . . 11 (𝐷 A → (x (A ∩ (𝑅 “ {𝐷})) ↔ (x A x𝑅𝐷)))
2524anbi1d 438 . . . . . . . . . 10 (𝐷 A → ((x (A ∩ (𝑅 “ {𝐷})) x𝐻y) ↔ ((x A x𝑅𝐷) x𝐻y)))
26 anass 381 . . . . . . . . . 10 (((x A x𝑅𝐷) x𝐻y) ↔ (x A (x𝑅𝐷 x𝐻y)))
2725, 26syl6bb 185 . . . . . . . . 9 (𝐷 A → ((x (A ∩ (𝑅 “ {𝐷})) x𝐻y) ↔ (x A (x𝑅𝐷 x𝐻y))))
2827adantl 262 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → ((x (A ∩ (𝑅 “ {𝐷})) x𝐻y) ↔ (x A (x𝑅𝐷 x𝐻y))))
29 isorel 5391 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (A, B) (x A 𝐷 A)) → (x𝑅𝐷 ↔ (𝐻x)𝑆(𝐻𝐷)))
30 fnbrfvb 5157 . . . . . . . . . . . . . . . . 17 ((𝐻 Fn A x A) → ((𝐻x) = yx𝐻y))
3130bicomd 129 . . . . . . . . . . . . . . . 16 ((𝐻 Fn A x A) → (x𝐻y ↔ (𝐻x) = y))
3212, 31sylan 267 . . . . . . . . . . . . . . 15 ((𝐻 Isom 𝑅, 𝑆 (A, B) x A) → (x𝐻y ↔ (𝐻x) = y))
3332adantrr 448 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (A, B) (x A 𝐷 A)) → (x𝐻y ↔ (𝐻x) = y))
3429, 33anbi12d 442 . . . . . . . . . . . . 13 ((𝐻 Isom 𝑅, 𝑆 (A, B) (x A 𝐷 A)) → ((x𝑅𝐷 x𝐻y) ↔ ((𝐻x)𝑆(𝐻𝐷) (𝐻x) = y)))
35 ancom 253 . . . . . . . . . . . . . 14 (((𝐻x)𝑆(𝐻𝐷) (𝐻x) = y) ↔ ((𝐻x) = y (𝐻x)𝑆(𝐻𝐷)))
36 breq1 3758 . . . . . . . . . . . . . . 15 ((𝐻x) = y → ((𝐻x)𝑆(𝐻𝐷) ↔ y𝑆(𝐻𝐷)))
3736pm5.32i 427 . . . . . . . . . . . . . 14 (((𝐻x) = y (𝐻x)𝑆(𝐻𝐷)) ↔ ((𝐻x) = y y𝑆(𝐻𝐷)))
3835, 37bitri 173 . . . . . . . . . . . . 13 (((𝐻x)𝑆(𝐻𝐷) (𝐻x) = y) ↔ ((𝐻x) = y y𝑆(𝐻𝐷)))
3934, 38syl6bb 185 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (A, B) (x A 𝐷 A)) → ((x𝑅𝐷 x𝐻y) ↔ ((𝐻x) = y y𝑆(𝐻𝐷))))
4039exp32 347 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (A, B) → (x A → (𝐷 A → ((x𝑅𝐷 x𝐻y) ↔ ((𝐻x) = y y𝑆(𝐻𝐷))))))
4140com23 72 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝐷 A → (x A → ((x𝑅𝐷 x𝐻y) ↔ ((𝐻x) = y y𝑆(𝐻𝐷))))))
4241imp 115 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (x A → ((x𝑅𝐷 x𝐻y) ↔ ((𝐻x) = y y𝑆(𝐻𝐷)))))
4342pm5.32d 423 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → ((x A (x𝑅𝐷 x𝐻y)) ↔ (x A ((𝐻x) = y y𝑆(𝐻𝐷)))))
4428, 43bitrd 177 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → ((x (A ∩ (𝑅 “ {𝐷})) x𝐻y) ↔ (x A ((𝐻x) = y y𝑆(𝐻𝐷)))))
4544rexbidv2 2323 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (x (A ∩ (𝑅 “ {𝐷}))x𝐻yx A ((𝐻x) = y y𝑆(𝐻𝐷))))
46 r19.41v 2460 . . . . . 6 (x A ((𝐻x) = y y𝑆(𝐻𝐷)) ↔ (x A (𝐻x) = y y𝑆(𝐻𝐷)))
4745, 46syl6bb 185 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (x (A ∩ (𝑅 “ {𝐷}))x𝐻y ↔ (x A (𝐻x) = y y𝑆(𝐻𝐷))))
4819, 47bitr4d 180 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → ((y B y (𝑆 “ {(𝐻𝐷)})) ↔ x (A ∩ (𝑅 “ {𝐷}))x𝐻y))
491, 48syl5bb 181 . . 3 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (y (B ∩ (𝑆 “ {(𝐻𝐷)})) ↔ x (A ∩ (𝑅 “ {𝐷}))x𝐻y))
5049abbi2dv 2153 . 2 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (B ∩ (𝑆 “ {(𝐻𝐷)})) = {yx (A ∩ (𝑅 “ {𝐷}))x𝐻y})
51 dfima2 4613 . 2 (𝐻 “ (A ∩ (𝑅 “ {𝐷}))) = {yx (A ∩ (𝑅 “ {𝐷}))x𝐻y}
5250, 51syl6reqr 2088 1 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (𝐻 “ (A ∩ (𝑅 “ {𝐷}))) = (B ∩ (𝑆 “ {(𝐻𝐷)})))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  Vcvv 2551  cin 2910  {csn 3367   class class class wbr 3755  ccnv 4287  ran crn 4289  cima 4291   Fn wfn 4840  ontowfo 4843  1-1-ontowf1o 4844  cfv 4845   Isom wiso 4846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-isom 4854
This theorem is referenced by:  isoini2  5401  isoselem  5402
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