Theorem isores1 5944

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
isores1	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))

Proof of Theorem isores1

Step	Hyp	Ref	Expression
1		isocnv 5941	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2		isores2 5943	. . . . 5 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) ↔ ◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
3	1, 2	sylib 122	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
4		isocnv 5941	. . . 4 ⊢ (◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴) → ◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
5	3, 4	syl 14	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
6		isof1o 5937	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
7		f1orel 5577	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Rel 𝐻)
8		dfrel2 5179	. . . . 5 ⊢ (Rel 𝐻 ↔ ◡◡𝐻 = 𝐻)
9		isoeq1 5931	. . . . 5 ⊢ (◡◡𝐻 = 𝐻 → (◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)))
10	8, 9	sylbi 121	. . . 4 ⊢ (Rel 𝐻 → (◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)))
11	6, 7, 10	3syl 17	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)))
12	5, 11	mpbid 147	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
13		isocnv 5941	. . . . 5 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → ◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
14	13, 2	sylibr 134	. . . 4 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
15		isocnv 5941	. . . 4 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
16	14, 15	syl 14	. . 3 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → ◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
17		isof1o 5937	. . . 4 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
18		isoeq1 5931	. . . . 5 ⊢ (◡◡𝐻 = 𝐻 → (◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
19	8, 18	sylbi 121	. . . 4 ⊢ (Rel 𝐻 → (◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
20	17, 7, 19	3syl 17	. . 3 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → (◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
21	16, 20	mpbid 147	. 2 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
22	12, 21	impbii 126	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))

Syntax hints:   ↔ wb 105   = wceq 1395   ∩ cin 3196   × cxp 4717  ^◡ccnv 4718  Rel wrel 4724  –1-1-onto→wf1o 5317   Isom wiso 5319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327

This theorem is referenced by: (None)