Theorem isores2 5494

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.

Step	Hyp	Ref	Expression
1		f1of 5288	. . . . . . . 8 ⊢ (H:A–1-1-onto→B → H:A–→B)
2		ffvelrn 5416	. . . . . . . . . 10 ⊢ ((H:A–→B ∧ x ∈ A) → (H ‘x) ∈ B)
3	2	adantrr 697	. . . . . . . . 9 ⊢ ((H:A–→B ∧ (x ∈ A ∧ y ∈ A)) → (H ‘x) ∈ B)
4		ffvelrn 5416	. . . . . . . . . 10 ⊢ ((H:A–→B ∧ y ∈ A) → (H ‘y) ∈ B)
5	4	adantrl 696	. . . . . . . . 9 ⊢ ((H:A–→B ∧ (x ∈ A ∧ y ∈ A)) → (H ‘y) ∈ B)
6		brinxp 4837	. . . . . . . . 9 ⊢ (((H ‘x) ∈ B ∧ (H ‘y) ∈ B) → ((H ‘x)S(H ‘y) ↔ (H ‘x)(S ∩ (B × B))(H ‘y)))
7	3, 5, 6	syl2anc 642	. . . . . . . 8 ⊢ ((H:A–→B ∧ (x ∈ A ∧ y ∈ A)) → ((H ‘x)S(H ‘y) ↔ (H ‘x)(S ∩ (B × B))(H ‘y)))
8	1, 7	sylan 457	. . . . . . 7 ⊢ ((H:A–1-1-onto→B ∧ (x ∈ A ∧ y ∈ A)) → ((H ‘x)S(H ‘y) ↔ (H ‘x)(S ∩ (B × B))(H ‘y)))
9	8	anassrs 629	. . . . . 6 ⊢ (((H:A–1-1-onto→B ∧ x ∈ A) ∧ y ∈ A) → ((H ‘x)S(H ‘y) ↔ (H ‘x)(S ∩ (B × B))(H ‘y)))
10	9	bibi2d 309	. . . . 5 ⊢ (((H:A–1-1-onto→B ∧ x ∈ A) ∧ y ∈ A) → ((xRy ↔ (H ‘x)S(H ‘y)) ↔ (xRy ↔ (H ‘x)(S ∩ (B × B))(H ‘y))))
11	10	ralbidva 2631	. . . 4 ⊢ ((H:A–1-1-onto→B ∧ x ∈ A) → (∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀y ∈ A (xRy ↔ (H ‘x)(S ∩ (B × B))(H ‘y))))
12	11	ralbidva 2631	. . 3 ⊢ (H:A–1-1-onto→B → (∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)(S ∩ (B × B))(H ‘y))))
13	12	pm5.32i 618	. 2 ⊢ ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)(S ∩ (B × B))(H ‘y))))
14		df-iso 4797	. 2 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
15		df-iso 4797	. 2 ⊢ (H Isom R, (S ∩ (B × B))(A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)(S ∩ (B × B))(H ‘y))))
16	13, 14, 15	3bitr4i 268	1 ⊢ (H Isom R, S (A, B) ↔ H Isom R, (S ∩ (B × B))(A, B))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2615   ∩ cin 3209   class class class wbr 4640   × cxp 4771  –→wf 4778  –1-1-onto→wf1o 4781   ‘cfv 4782   Isom wiso 4783

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-f1o 4795  df-fv 4796  df-iso 4797

This theorem is referenced by:  isores1  5495