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Theorem isomin 5496
 Description: Isomorphisms preserve minimal elements. Note that (◡R “ {D}) is Takeuti and Zaring's idiom for the initial segment {x ∣ xRD}. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by set.mm contributors, 19-Apr-2004.)
Assertion
Ref Expression
isomin ((H Isom R, S (A, B) (C A D A)) → ((C ∩ (R “ {D})) = ↔ ((HC) ∩ (S “ {(HD)})) = ))

Proof of Theorem isomin
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel2 3268 . . . . . . . 8 ((C A y C) → y A)
21anim1i 551 . . . . . . 7 (((C A y C) D A) → (y A D A))
32an32s 779 . . . . . 6 (((C A D A) y C) → (y A D A))
4 isorel 5489 . . . . . . 7 ((H Isom R, S (A, B) (y A D A)) → (yRD ↔ (Hy)S(HD)))
5 fvex 5339 . . . . . . . . 9 (Hy) V
6 breq1 4642 . . . . . . . . 9 (x = (Hy) → (xS(HD) ↔ (Hy)S(HD)))
75, 6ceqsexv 2894 . . . . . . . 8 (x(x = (Hy) xS(HD)) ↔ (Hy)S(HD))
8 eqcom 2355 . . . . . . . . . . 11 (x = (Hy) ↔ (Hy) = x)
9 isof1o 5488 . . . . . . . . . . . . 13 (H Isom R, S (A, B) → H:A1-1-ontoB)
10 f1ofn 5288 . . . . . . . . . . . . 13 (H:A1-1-ontoBH Fn A)
119, 10syl 15 . . . . . . . . . . . 12 (H Isom R, S (A, B) → H Fn A)
12 simpl 443 . . . . . . . . . . . 12 ((y A D A) → y A)
13 fnbrfvb 5358 . . . . . . . . . . . 12 ((H Fn A y A) → ((Hy) = xyHx))
1411, 12, 13syl2an 463 . . . . . . . . . . 11 ((H Isom R, S (A, B) (y A D A)) → ((Hy) = xyHx))
158, 14syl5bb 248 . . . . . . . . . 10 ((H Isom R, S (A, B) (y A D A)) → (x = (Hy) ↔ yHx))
1615anbi1d 685 . . . . . . . . 9 ((H Isom R, S (A, B) (y A D A)) → ((x = (Hy) xS(HD)) ↔ (yHx xS(HD))))
1716exbidv 1626 . . . . . . . 8 ((H Isom R, S (A, B) (y A D A)) → (x(x = (Hy) xS(HD)) ↔ x(yHx xS(HD))))
187, 17syl5bbr 250 . . . . . . 7 ((H Isom R, S (A, B) (y A D A)) → ((Hy)S(HD) ↔ x(yHx xS(HD))))
194, 18bitrd 244 . . . . . 6 ((H Isom R, S (A, B) (y A D A)) → (yRDx(yHx xS(HD))))
203, 19sylan2 460 . . . . 5 ((H Isom R, S (A, B) ((C A D A) y C)) → (yRDx(yHx xS(HD))))
2120anassrs 629 . . . 4 (((H Isom R, S (A, B) (C A D A)) y C) → (yRDx(yHx xS(HD))))
2221rexbidva 2631 . . 3 ((H Isom R, S (A, B) (C A D A)) → (y C yRDy C x(yHx xS(HD))))
23 elin 3219 . . . . . 6 (y (C ∩ (R “ {D})) ↔ (y C y (R “ {D})))
24 eliniseg 5020 . . . . . . 7 (y (R “ {D}) ↔ yRD)
2524anbi2i 675 . . . . . 6 ((y C y (R “ {D})) ↔ (y C yRD))
2623, 25bitri 240 . . . . 5 (y (C ∩ (R “ {D})) ↔ (y C yRD))
2726exbii 1582 . . . 4 (y y (C ∩ (R “ {D})) ↔ y(y C yRD))
28 neq0 3560 . . . 4 (¬ (C ∩ (R “ {D})) = y y (C ∩ (R “ {D})))
29 df-rex 2620 . . . 4 (y C yRDy(y C yRD))
3027, 28, 293bitr4i 268 . . 3 (¬ (C ∩ (R “ {D})) = y C yRD)
31 elima 4754 . . . . . . 7 (x (HC) ↔ y C yHx)
32 eliniseg 5020 . . . . . . 7 (x (S “ {(HD)}) ↔ xS(HD))
3331, 32anbi12i 678 . . . . . 6 ((x (HC) x (S “ {(HD)})) ↔ (y C yHx xS(HD)))
34 elin 3219 . . . . . 6 (x ((HC) ∩ (S “ {(HD)})) ↔ (x (HC) x (S “ {(HD)})))
35 r19.41v 2764 . . . . . 6 (y C (yHx xS(HD)) ↔ (y C yHx xS(HD)))
3633, 34, 353bitr4i 268 . . . . 5 (x ((HC) ∩ (S “ {(HD)})) ↔ y C (yHx xS(HD)))
3736exbii 1582 . . . 4 (x x ((HC) ∩ (S “ {(HD)})) ↔ xy C (yHx xS(HD)))
38 neq0 3560 . . . 4 (¬ ((HC) ∩ (S “ {(HD)})) = x x ((HC) ∩ (S “ {(HD)})))
39 rexcom4 2878 . . . 4 (y C x(yHx xS(HD)) ↔ xy C (yHx xS(HD)))
4037, 38, 393bitr4i 268 . . 3 (¬ ((HC) ∩ (S “ {(HD)})) = y C x(yHx xS(HD)))
4122, 30, 403bitr4g 279 . 2 ((H Isom R, S (A, B) (C A D A)) → (¬ (C ∩ (R “ {D})) = ↔ ¬ ((HC) ∩ (S “ {(HD)})) = ))
4241con4bid 284 1 ((H Isom R, S (A, B) (C A D A)) → ((C ∩ (R “ {D})) = ↔ ((HC) ∩ (S “ {(HD)})) = ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  {csn 3737   class class class wbr 4639   “ cima 4722  ◡ccnv 4771   Fn wfn 4776  –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-f1o 4794  df-fv 4795  df-iso 4796 This theorem is referenced by: (None)
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