Theorem isores2 7353

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 6848	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
2		ffvelcdm 7101	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
3	2	adantrr 717	. . . . . . . . 9 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑥) ∈ 𝐵)
4		ffvelcdm 7101	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ 𝐵)
5	4	adantrl 716	. . . . . . . . 9 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑦) ∈ 𝐵)
6		brinxp 5764	. . . . . . . . 9 ⊢ (((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻‘𝑦)))
7	3, 5, 6	syl2anc 584	. . . . . . . 8 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻‘𝑦)))
8	1, 7	sylan 580	. . . . . . 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 467	. . . . . 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 342	. . . . 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 3176	. . . 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 3176	. . 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 574	. 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-isom 6570	. 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-isom 6570	. 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 303	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3061   ∩ cin 3950   class class class wbr 5143   × cxp 5683  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561   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-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-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-f1o 6568  df-fv 6569  df-isom 6570

This theorem is referenced by:  isores1  7354  hartogslem1  9582  leiso  14498  icopnfhmeo  24974  iccpnfhmeo  24976  gtiso  32710  xrge0iifhmeo  33935