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Theorem f1oiso 5499
 Description: Any one-to-one onto function determines an isomorphism with an induced relation S. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by set.mm contributors, 30-Apr-2004.)
Assertion
Ref Expression
f1oiso ((H:A1-1-ontoB S = {z, w x A y A ((z = (Hx) w = (Hy)) xRy)}) → H Isom R, S (A, B))
Distinct variable groups:   x,y,z,w,A   x,B,y   x,H,y,z,w   x,R,y,z,w
Allowed substitution hints:   B(z,w)   S(x,y,z,w)

Proof of Theorem f1oiso
Dummy variables v u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . 2 ((H:A1-1-ontoB S = {z, w x A y A ((z = (Hx) w = (Hy)) xRy)}) → H:A1-1-ontoB)
2 f1of1 5286 . . 3 (H:A1-1-ontoBH:A1-1B)
3 df-br 4640 . . . . 5 ((Hv)S(Hu) ↔ (Hv), (Hu) S)
4 eleq2 2414 . . . . . . 7 (S = {z, w x A y A ((z = (Hx) w = (Hy)) xRy)} → ((Hv), (Hu) S(Hv), (Hu) {z, w x A y A ((z = (Hx) w = (Hy)) xRy)}))
5 fvex 5339 . . . . . . . . 9 (Hv) V
6 fvex 5339 . . . . . . . . 9 (Hu) V
7 eqeq1 2359 . . . . . . . . . . . 12 (z = (Hv) → (z = (Hx) ↔ (Hv) = (Hx)))
87anbi1d 685 . . . . . . . . . . 11 (z = (Hv) → ((z = (Hx) w = (Hy)) ↔ ((Hv) = (Hx) w = (Hy))))
98anbi1d 685 . . . . . . . . . 10 (z = (Hv) → (((z = (Hx) w = (Hy)) xRy) ↔ (((Hv) = (Hx) w = (Hy)) xRy)))
1092rexbidv 2657 . . . . . . . . 9 (z = (Hv) → (x A y A ((z = (Hx) w = (Hy)) xRy) ↔ x A y A (((Hv) = (Hx) w = (Hy)) xRy)))
11 eqeq1 2359 . . . . . . . . . . . 12 (w = (Hu) → (w = (Hy) ↔ (Hu) = (Hy)))
1211anbi2d 684 . . . . . . . . . . 11 (w = (Hu) → (((Hv) = (Hx) w = (Hy)) ↔ ((Hv) = (Hx) (Hu) = (Hy))))
1312anbi1d 685 . . . . . . . . . 10 (w = (Hu) → ((((Hv) = (Hx) w = (Hy)) xRy) ↔ (((Hv) = (Hx) (Hu) = (Hy)) xRy)))
14132rexbidv 2657 . . . . . . . . 9 (w = (Hu) → (x A y A (((Hv) = (Hx) w = (Hy)) xRy) ↔ x A y A (((Hv) = (Hx) (Hu) = (Hy)) xRy)))
155, 6, 10, 14opelopab 4708 . . . . . . . 8 ((Hv), (Hu) {z, w x A y A ((z = (Hx) w = (Hy)) xRy)} ↔ x A y A (((Hv) = (Hx) (Hu) = (Hy)) xRy))
16 anass 630 . . . . . . . . . . . . . . 15 ((((Hv) = (Hx) (Hu) = (Hy)) xRy) ↔ ((Hv) = (Hx) ((Hu) = (Hy) xRy)))
17 f1fveq 5473 . . . . . . . . . . . . . . . . . 18 ((H:A1-1B (v A x A)) → ((Hv) = (Hx) ↔ v = x))
18 eqcom 2355 . . . . . . . . . . . . . . . . . 18 (v = xx = v)
1917, 18syl6bb 252 . . . . . . . . . . . . . . . . 17 ((H:A1-1B (v A x A)) → ((Hv) = (Hx) ↔ x = v))
2019anassrs 629 . . . . . . . . . . . . . . . 16 (((H:A1-1B v A) x A) → ((Hv) = (Hx) ↔ x = v))
2120anbi1d 685 . . . . . . . . . . . . . . 15 (((H:A1-1B v A) x A) → (((Hv) = (Hx) ((Hu) = (Hy) xRy)) ↔ (x = v ((Hu) = (Hy) xRy))))
2216, 21syl5bb 248 . . . . . . . . . . . . . 14 (((H:A1-1B v A) x A) → ((((Hv) = (Hx) (Hu) = (Hy)) xRy) ↔ (x = v ((Hu) = (Hy) xRy))))
2322rexbidv 2635 . . . . . . . . . . . . 13 (((H:A1-1B v A) x A) → (y A (((Hv) = (Hx) (Hu) = (Hy)) xRy) ↔ y A (x = v ((Hu) = (Hy) xRy))))
24 r19.42v 2765 . . . . . . . . . . . . 13 (y A (x = v ((Hu) = (Hy) xRy)) ↔ (x = v y A ((Hu) = (Hy) xRy)))
2523, 24syl6bb 252 . . . . . . . . . . . 12 (((H:A1-1B v A) x A) → (y A (((Hv) = (Hx) (Hu) = (Hy)) xRy) ↔ (x = v y A ((Hu) = (Hy) xRy))))
2625rexbidva 2631 . . . . . . . . . . 11 ((H:A1-1B v A) → (x A y A (((Hv) = (Hx) (Hu) = (Hy)) xRy) ↔ x A (x = v y A ((Hu) = (Hy) xRy))))
27 breq1 4642 . . . . . . . . . . . . . . 15 (x = v → (xRyvRy))
2827anbi2d 684 . . . . . . . . . . . . . 14 (x = v → (((Hu) = (Hy) xRy) ↔ ((Hu) = (Hy) vRy)))
2928rexbidv 2635 . . . . . . . . . . . . 13 (x = v → (y A ((Hu) = (Hy) xRy) ↔ y A ((Hu) = (Hy) vRy)))
3029ceqsrexv 2972 . . . . . . . . . . . 12 (v A → (x A (x = v y A ((Hu) = (Hy) xRy)) ↔ y A ((Hu) = (Hy) vRy)))
3130adantl 452 . . . . . . . . . . 11 ((H:A1-1B v A) → (x A (x = v y A ((Hu) = (Hy) xRy)) ↔ y A ((Hu) = (Hy) vRy)))
3226, 31bitrd 244 . . . . . . . . . 10 ((H:A1-1B v A) → (x A y A (((Hv) = (Hx) (Hu) = (Hy)) xRy) ↔ y A ((Hu) = (Hy) vRy)))
33 f1fveq 5473 . . . . . . . . . . . . . . 15 ((H:A1-1B (u A y A)) → ((Hu) = (Hy) ↔ u = y))
34 eqcom 2355 . . . . . . . . . . . . . . 15 (u = yy = u)
3533, 34syl6bb 252 . . . . . . . . . . . . . 14 ((H:A1-1B (u A y A)) → ((Hu) = (Hy) ↔ y = u))
3635anassrs 629 . . . . . . . . . . . . 13 (((H:A1-1B u A) y A) → ((Hu) = (Hy) ↔ y = u))
3736anbi1d 685 . . . . . . . . . . . 12 (((H:A1-1B u A) y A) → (((Hu) = (Hy) vRy) ↔ (y = u vRy)))
3837rexbidva 2631 . . . . . . . . . . 11 ((H:A1-1B u A) → (y A ((Hu) = (Hy) vRy) ↔ y A (y = u vRy)))
39 breq2 4643 . . . . . . . . . . . . 13 (y = u → (vRyvRu))
4039ceqsrexv 2972 . . . . . . . . . . . 12 (u A → (y A (y = u vRy) ↔ vRu))
4140adantl 452 . . . . . . . . . . 11 ((H:A1-1B u A) → (y A (y = u vRy) ↔ vRu))
4238, 41bitrd 244 . . . . . . . . . 10 ((H:A1-1B u A) → (y A ((Hu) = (Hy) vRy) ↔ vRu))
4332, 42sylan9bb 680 . . . . . . . . 9 (((H:A1-1B v A) (H:A1-1B u A)) → (x A y A (((Hv) = (Hx) (Hu) = (Hy)) xRy) ↔ vRu))
4443anandis 803 . . . . . . . 8 ((H:A1-1B (v A u A)) → (x A y A (((Hv) = (Hx) (Hu) = (Hy)) xRy) ↔ vRu))
4515, 44syl5bb 248 . . . . . . 7 ((H:A1-1B (v A u A)) → ((Hv), (Hu) {z, w x A y A ((z = (Hx) w = (Hy)) xRy)} ↔ vRu))
464, 45sylan9bbr 681 . . . . . 6 (((H:A1-1B (v A u A)) S = {z, w x A y A ((z = (Hx) w = (Hy)) xRy)}) → ((Hv), (Hu) SvRu))
4746an32s 779 . . . . 5 (((H:A1-1B S = {z, w x A y A ((z = (Hx) w = (Hy)) xRy)}) (v A u A)) → ((Hv), (Hu) SvRu))
483, 47syl5rbb 249 . . . 4 (((H:A1-1B S = {z, w x A y A ((z = (Hx) w = (Hy)) xRy)}) (v A u A)) → (vRu ↔ (Hv)S(Hu)))
4948ralrimivva 2706 . . 3 ((H:A1-1B S = {z, w x A y A ((z = (Hx) w = (Hy)) xRy)}) → v A u A (vRu ↔ (Hv)S(Hu)))
502, 49sylan 457 . 2 ((H:A1-1-ontoB S = {z, w x A y A ((z = (Hx) w = (Hy)) xRy)}) → v A u A (vRu ↔ (Hv)S(Hu)))
51 df-iso 4796 . 2 (H Isom R, S (A, B) ↔ (H:A1-1-ontoB v A u A (vRu ↔ (Hv)S(Hu))))
521, 50, 51sylanbrc 645 1 ((H:A1-1-ontoB S = {z, w x A y A ((z = (Hx) w = (Hy)) xRy)}) → H Isom R, S (A, B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  ⟨cop 4561  {copab 4622   class class class wbr 4639  –1-1→wf1 4778  –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-cnv 4785  df-rn 4786  df-dm 4787  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:  f1oiso2  5500
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