Theorem isores2 5992

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 5619	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
2		ffvelcdm 5815	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
3	2	adantrr 479	. . . . . . . . 9 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑥) ∈ 𝐵)
4		ffvelcdm 5815	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ 𝐵)
5	4	adantrl 478	. . . . . . . . 9 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑦) ∈ 𝐵)
6		brinxp 4823	. . . . . . . . 9 ⊢ (((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻‘𝑦)))
7	3, 5, 6	syl2anc 411	. . . . . . . 8 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑥)(𝑆 ∩ (𝐵 × 𝐵))(𝐻‘𝑦)))
8	1, 7	sylan 283	. . . . . . 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 400	. . . . . 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 232	. . . . 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 2540	. . . 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 2540	. . 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 454	. 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 5366	. 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 5366	. 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 212	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2205  ∀wral 2522   ∩ cin 3213   class class class wbr 4114   × cxp 4752  ⟶wf 5353  –1-1-onto→wf1o 5356  ‘cfv 5357   Isom wiso 5358

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-f1o 5364  df-fv 5365  df-isom 5366

This theorem is referenced by:  isores1  5993