Theorem isores1 5883

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 5880	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2		isores2 5882	. . . . 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 5880	. . . 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 5876	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
7		f1orel 5525	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Rel 𝐻)
8		dfrel2 5133	. . . . 5 ⊢ (Rel 𝐻 ↔ ◡◡𝐻 = 𝐻)
9		isoeq1 5870	. . . . 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 5880	. . . . 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 5880	. . . 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 5876	. . . 4 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
18		isoeq1 5870	. . . . 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 1373   ∩ cin 3165   × cxp 4673  ^◡ccnv 4674  Rel wrel 4680  –1-1-onto→wf1o 5270   Isom wiso 5272

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280

This theorem is referenced by: (None)