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Theorem isoini 5378
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 3099 . . . 4 (y (B ∩ (𝑆 “ {(𝐻𝐷)})) ↔ (y B y (𝑆 “ {(𝐻𝐷)})))
2 isof1o 5368 . . . . . . . . 9 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A1-1-ontoB)
3 f1ofo 5054 . . . . . . . . 9 (𝐻:A1-1-ontoB𝐻:AontoB)
4 forn 5030 . . . . . . . . . 10 (𝐻:AontoB → ran 𝐻 = B)
54eleq2d 2085 . . . . . . . . 9 (𝐻:AontoB → (y ran 𝐻y B))
62, 3, 53syl 17 . . . . . . . 8 (𝐻 Isom 𝑅, 𝑆 (A, B) → (y ran 𝐻y B))
7 f1ofn 5048 . . . . . . . . 9 (𝐻:A1-1-ontoB𝐻 Fn A)
8 fvelrnb 5142 . . . . . . . . 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 5113 . . . . . . . 8 ((Fun 𝐻 𝐷 dom 𝐻) → (𝐻𝐷) V)
1514funfni 4921 . . . . . . 7 ((𝐻 Fn A 𝐷 A) → (𝐻𝐷) V)
16 vex 2534 . . . . . . . 8 y V
1716eliniseg 4618 . . . . . . 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 445 . . . . 5 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → ((y B y (𝑆 “ {(𝐻𝐷)})) ↔ (x A (𝐻x) = y y𝑆(𝐻𝐷))))
20 elin 3099 . . . . . . . . . . . 12 (x (A ∩ (𝑅 “ {𝐷})) ↔ (x A x (𝑅 “ {𝐷})))
21 vex 2534 . . . . . . . . . . . . . 14 x V
2221eliniseg 4618 . . . . . . . . . . . . 13 (𝐷 A → (x (𝑅 “ {𝐷}) ↔ x𝑅𝐷))
2322anbi2d 440 . . . . . . . . . . . 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 441 . . . . . . . . . 10 (𝐷 A → ((x (A ∩ (𝑅 “ {𝐷})) x𝐻y) ↔ ((x A x𝑅𝐷) x𝐻y)))
26 anass 383 . . . . . . . . . 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 5369 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (A, B) (x A 𝐷 A)) → (x𝑅𝐷 ↔ (𝐻x)𝑆(𝐻𝐷)))
30 fnbrfvb 5135 . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 14 ((𝐻 Isom 𝑅, 𝑆 (A, B) (x A 𝐷 A)) → (x𝐻y ↔ (𝐻x) = y))
3429, 33anbi12d 445 . . . . . . . . . . . . 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 3737 . . . . . . . . . . . . . . 15 ((𝐻x) = y → ((𝐻x)𝑆(𝐻𝐷) ↔ y𝑆(𝐻𝐷)))
3736pm5.32i 430 . . . . . . . . . . . . . 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 426 . . . . . . . 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 2303 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (x (A ∩ (𝑅 “ {𝐷}))x𝐻yx A ((𝐻x) = y y𝑆(𝐻𝐷))))
46 r19.41v 2440 . . . . . 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 2134 . 2 ((𝐻 Isom 𝑅, 𝑆 (A, B) 𝐷 A) → (B ∩ (𝑆 “ {(𝐻𝐷)})) = {yx (A ∩ (𝑅 “ {𝐷}))x𝐻y})
51 dfima2 4593 . 2 (𝐻 “ (A ∩ (𝑅 “ {𝐷}))) = {yx (A ∩ (𝑅 “ {𝐷}))x𝐻y}
5250, 51syl6reqr 2069 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 1226   wcel 1370  {cab 2004  wrex 2281  Vcvv 2531  cin 2889  {csn 3346   class class class wbr 3734  ccnv 4267  ran crn 4269  cima 4271   Fn wfn 4820  ontowfo 4823  1-1-ontowf1o 4824  cfv 4825   Isom wiso 4826
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-isom 4834
This theorem is referenced by:  isoini2  5379  isoselem  5380
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