Theorem isores1 6840

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 6836	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2		isores2 6839	. . . . 5 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) ↔ ◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
3	1, 2	sylib 210	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
4		isocnv 6836	. . . 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 6829	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
7		f1orel 6382	. . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Rel 𝐻)
8		dfrel2 5825	. . . . 5 ⊢ (Rel 𝐻 ↔ ◡◡𝐻 = 𝐻)
9		isoeq1 6823	. . . . 5 ⊢ (◡◡𝐻 = 𝐻 → (◡◡𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)))
10	8, 9	sylbi 209	. . . 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 224	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
13		isocnv 6836	. . . . 5 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → ◡𝐻 Isom 𝑆, (𝑅 ∩ (𝐴 × 𝐴))(𝐵, 𝐴))
14	13, 2	sylibr 226	. . . 4 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴))
15		isocnv 6836	. . . 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 6829	. . . 4 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
18		isoeq1 6823	. . . . 5 ⊢ (◡◡𝐻 = 𝐻 → (◡◡𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
19	8, 18	sylbi 209	. . . 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 224	. 2 ⊢ (𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
22	12, 21	impbii 201	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))

Syntax hints:   ↔ wb 198   = wceq 1658   ∩ cin 3798   × cxp 5341  ^◡ccnv 5342  Rel wrel 5348  –1-1-onto→wf1o 6123   Isom wiso 6125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-isom 6133

This theorem is referenced by:  leiso  13533  icopnfhmeo  23113  iccpnfhmeo  23115  xrhmeo  23116  gtiso  30027  xrge0iifhmeo  30528