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Theorem isoini 5497
 Description: Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by set.mm contributors, 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 3219 . . . . 5 (y (B ∩ (S “ {(HD)})) ↔ (y B y (S “ {(HD)})))
2 eliniseg 5020 . . . . . 6 (y (S “ {(HD)}) ↔ yS(HD))
32anbi2i 675 . . . . 5 ((y B y (S “ {(HD)})) ↔ (y B yS(HD)))
41, 3bitri 240 . . . 4 (y (B ∩ (S “ {(HD)})) ↔ (y B yS(HD)))
5 isof1o 5488 . . . . . . . . . . 11 (H Isom R, S (A, B) → H:A1-1-ontoB)
6 f1ofo 5293 . . . . . . . . . . 11 (H:A1-1-ontoBH:AontoB)
75, 6syl 15 . . . . . . . . . 10 (H Isom R, S (A, B) → H:AontoB)
8 forn 5272 . . . . . . . . . 10 (H:AontoB → ran H = B)
97, 8syl 15 . . . . . . . . 9 (H Isom R, S (A, B) → ran H = B)
109eleq2d 2420 . . . . . . . 8 (H Isom R, S (A, B) → (y ran Hy B))
11 f1ofn 5288 . . . . . . . . . 10 (H:A1-1-ontoBH Fn A)
125, 11syl 15 . . . . . . . . 9 (H Isom R, S (A, B) → H Fn A)
13 fvelrnb 5365 . . . . . . . . 9 (H Fn A → (y ran Hx A (Hx) = y))
1412, 13syl 15 . . . . . . . 8 (H Isom R, S (A, B) → (y ran Hx A (Hx) = y))
1510, 14bitr3d 246 . . . . . . 7 (H Isom R, S (A, B) → (y Bx A (Hx) = y))
1615anbi1d 685 . . . . . 6 (H Isom R, S (A, B) → ((y B yS(HD)) ↔ (x A (Hx) = y yS(HD))))
1716adantr 451 . . . . 5 ((H Isom R, S (A, B) D A) → ((y B yS(HD)) ↔ (x A (Hx) = y yS(HD))))
18 elin 3219 . . . . . . . . . . 11 (x (A ∩ (R “ {D})) ↔ (x A x (R “ {D})))
19 eliniseg 5020 . . . . . . . . . . . 12 (x (R “ {D}) ↔ xRD)
2019anbi2i 675 . . . . . . . . . . 11 ((x A x (R “ {D})) ↔ (x A xRD))
2118, 20bitri 240 . . . . . . . . . 10 (x (A ∩ (R “ {D})) ↔ (x A xRD))
2221anbi1i 676 . . . . . . . . 9 ((x (A ∩ (R “ {D})) xHy) ↔ ((x A xRD) xHy))
23 anass 630 . . . . . . . . 9 (((x A xRD) xHy) ↔ (x A (xRD xHy)))
2422, 23bitri 240 . . . . . . . 8 ((x (A ∩ (R “ {D})) xHy) ↔ (x A (xRD xHy)))
25 fnbrfvb 5358 . . . . . . . . . . . . . . . . 17 ((H Fn A x A) → ((Hx) = yxHy))
2612, 25sylan 457 . . . . . . . . . . . . . . . 16 ((H Isom R, S (A, B) x A) → ((Hx) = yxHy))
2726adantrr 697 . . . . . . . . . . . . . . 15 ((H Isom R, S (A, B) (x A D A)) → ((Hx) = yxHy))
2827bicomd 192 . . . . . . . . . . . . . 14 ((H Isom R, S (A, B) (x A D A)) → (xHy ↔ (Hx) = y))
29 isorel 5489 . . . . . . . . . . . . . 14 ((H Isom R, S (A, B) (x A D A)) → (xRD ↔ (Hx)S(HD)))
3028, 29anbi12d 691 . . . . . . . . . . . . 13 ((H Isom R, S (A, B) (x A D A)) → ((xHy xRD) ↔ ((Hx) = y (Hx)S(HD))))
31 ancom 437 . . . . . . . . . . . . 13 ((xHy xRD) ↔ (xRD xHy))
32 breq1 4642 . . . . . . . . . . . . . 14 ((Hx) = y → ((Hx)S(HD) ↔ yS(HD)))
3332pm5.32i 618 . . . . . . . . . . . . 13 (((Hx) = y (Hx)S(HD)) ↔ ((Hx) = y yS(HD)))
3430, 31, 333bitr3g 278 . . . . . . . . . . . 12 ((H Isom R, S (A, B) (x A D A)) → ((xRD xHy) ↔ ((Hx) = y yS(HD))))
3534exp32 588 . . . . . . . . . . 11 (H Isom R, S (A, B) → (x A → (D A → ((xRD xHy) ↔ ((Hx) = y yS(HD))))))
3635com23 72 . . . . . . . . . 10 (H Isom R, S (A, B) → (D A → (x A → ((xRD xHy) ↔ ((Hx) = y yS(HD))))))
3736imp 418 . . . . . . . . 9 ((H Isom R, S (A, B) D A) → (x A → ((xRD xHy) ↔ ((Hx) = y yS(HD)))))
3837pm5.32d 620 . . . . . . . 8 ((H Isom R, S (A, B) D A) → ((x A (xRD xHy)) ↔ (x A ((Hx) = y yS(HD)))))
3924, 38syl5bb 248 . . . . . . 7 ((H Isom R, S (A, B) D A) → ((x (A ∩ (R “ {D})) xHy) ↔ (x A ((Hx) = y yS(HD)))))
4039rexbidv2 2637 . . . . . 6 ((H Isom R, S (A, B) D A) → (x (A ∩ (R “ {D}))xHyx A ((Hx) = y yS(HD))))
41 r19.41v 2764 . . . . . 6 (x A ((Hx) = y yS(HD)) ↔ (x A (Hx) = y yS(HD)))
4240, 41syl6bb 252 . . . . 5 ((H Isom R, S (A, B) D A) → (x (A ∩ (R “ {D}))xHy ↔ (x A (Hx) = y yS(HD))))
4317, 42bitr4d 247 . . . 4 ((H Isom R, S (A, B) D A) → ((y B yS(HD)) ↔ x (A ∩ (R “ {D}))xHy))
444, 43syl5bb 248 . . 3 ((H Isom R, S (A, B) D A) → (y (B ∩ (S “ {(HD)})) ↔ x (A ∩ (R “ {D}))xHy))
4544abbi2dv 2468 . 2 ((H Isom R, S (A, B) D A) → (B ∩ (S “ {(HD)})) = {y x (A ∩ (R “ {D}))xHy})
46 df-ima 4727 . 2 (H “ (A ∩ (R “ {D}))) = {y x (A ∩ (R “ {D}))xHy}
4745, 46syl6reqr 2404 1 ((H Isom R, S (A, B) D A) → (H “ (A ∩ (R “ {D}))) = (B ∩ (S “ {(HD)})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615   ∩ cin 3208  {csn 3737   class class class wbr 4639   “ cima 4722  ◡ccnv 4771  ran crn 4773   Fn wfn 4776  –onto→wfo 4779  –1-1-onto→wf1o 4780   ‘cfv 4781   Isom wiso 4782 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-iso 4796 This theorem is referenced by:  isoini2  5498
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