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Theorem isores2 5493
 Description: An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
isores2 (H Isom R, S (A, B) ↔ H Isom R, (S ∩ (B × B))(A, B))

Proof of Theorem isores2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1of 5287 . . . . . . . 8 (H:A1-1-ontoBH:A–→B)
2 ffvelrn 5415 . . . . . . . . . 10 ((H:A–→B x A) → (Hx) B)
32adantrr 697 . . . . . . . . 9 ((H:A–→B (x A y A)) → (Hx) B)
4 ffvelrn 5415 . . . . . . . . . 10 ((H:A–→B y A) → (Hy) B)
54adantrl 696 . . . . . . . . 9 ((H:A–→B (x A y A)) → (Hy) B)
6 brinxp 4836 . . . . . . . . 9 (((Hx) B (Hy) B) → ((Hx)S(Hy) ↔ (Hx)(S ∩ (B × B))(Hy)))
73, 5, 6syl2anc 642 . . . . . . . 8 ((H:A–→B (x A y A)) → ((Hx)S(Hy) ↔ (Hx)(S ∩ (B × B))(Hy)))
81, 7sylan 457 . . . . . . 7 ((H:A1-1-ontoB (x A y A)) → ((Hx)S(Hy) ↔ (Hx)(S ∩ (B × B))(Hy)))
98anassrs 629 . . . . . 6 (((H:A1-1-ontoB x A) y A) → ((Hx)S(Hy) ↔ (Hx)(S ∩ (B × B))(Hy)))
109bibi2d 309 . . . . 5 (((H:A1-1-ontoB x A) y A) → ((xRy ↔ (Hx)S(Hy)) ↔ (xRy ↔ (Hx)(S ∩ (B × B))(Hy))))
1110ralbidva 2630 . . . 4 ((H:A1-1-ontoB x A) → (y A (xRy ↔ (Hx)S(Hy)) ↔ y A (xRy ↔ (Hx)(S ∩ (B × B))(Hy))))
1211ralbidva 2630 . . 3 (H:A1-1-ontoB → (x A y A (xRy ↔ (Hx)S(Hy)) ↔ x A y A (xRy ↔ (Hx)(S ∩ (B × B))(Hy))))
1312pm5.32i 618 . 2 ((H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))) ↔ (H:A1-1-ontoB x A y A (xRy ↔ (Hx)(S ∩ (B × B))(Hy))))
14 df-iso 4796 . 2 (H Isom R, S (A, B) ↔ (H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))))
15 df-iso 4796 . 2 (H Isom R, (S ∩ (B × B))(A, B) ↔ (H:A1-1-ontoB x A y A (xRy ↔ (Hx)(S ∩ (B × B))(Hy))))
1613, 14, 153bitr4i 268 1 (H Isom R, S (A, B) ↔ H Isom R, (S ∩ (B × B))(A, B))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2614   ∩ cin 3208   class class class wbr 4639   × cxp 4770  –→wf 4777  –1-1-onto→wf1o 4780   ‘cfv 4781   Isom wiso 4782 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-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:  isores1  5494
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