Theorem isores1 7354

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 7350	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2		isores2 7353	. . . . 5 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) ↔ ◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
3	1, 2	sylib 218	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
4		isocnv 7350	. . . 4 ⊢ (◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴) → ◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
5	3, 4	syl 17	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
6		isof1o 7343	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
7		f1orel 6851	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Rel 𝐻)
8		dfrel2 6209	. . . . 5 ⊢ (Rel 𝐻 ↔ ◡◡𝐻 = 𝐻)
9		isoeq1 7337	. . . . 5 ⊢ (◡◡𝐻 = 𝐻 → (◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)))
10	8, 9	sylbi 217	. . . 4 ⊢ (Rel 𝐻 → (◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)))
11	6, 7, 10	3syl 18	. . 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 232	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
13		isocnv 7350	. . . . 5 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → ◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
14	13, 2	sylibr 234	. . . 4 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
15		isocnv 7350	. . . 4 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
16	14, 15	syl 17	. . 3 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → ◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
17		isof1o 7343	. . . 4 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
18		isoeq1 7337	. . . . 5 ⊢ (◡◡𝐻 = 𝐻 → (◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
19	8, 18	sylbi 217	. . . 4 ⊢ (Rel 𝐻 → (◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
20	17, 7, 19	3syl 18	. . 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 232	. 2 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
22	12, 21	impbii 209	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))

Syntax hints:   ↔ wb 206   = wceq 1540   ∩ cin 3950   × cxp 5683  ^◡ccnv 5684  Rel wrel 5690  –1-1-onto→wf1o 6560   Isom wiso 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570

This theorem is referenced by:  leiso  14498  icopnfhmeo  24974  iccpnfhmeo  24976  xrhmeo  24977  gtiso  32710  xrge0iifhmeo  33935