Theorem isosolem 6970

Description: Lemma for isoso 6971. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Assertion

Ref	Expression
isosolem	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

Proof of Theorem isosolem

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isopolem 6968	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))
2		isof1o 6946	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3		f1of 6490	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
4		ffvelrn 6721	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑐 ∈ 𝐴) → (𝐻‘𝑐) ∈ 𝐵)
5	4	ex 413	. . . . . . . . 9 ⊢ (𝐻:𝐴⟶𝐵 → (𝑐 ∈ 𝐴 → (𝐻‘𝑐) ∈ 𝐵))
6		ffvelrn 6721	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑑 ∈ 𝐴) → (𝐻‘𝑑) ∈ 𝐵)
7	6	ex 413	. . . . . . . . 9 ⊢ (𝐻:𝐴⟶𝐵 → (𝑑 ∈ 𝐴 → (𝐻‘𝑑) ∈ 𝐵))
8	5, 7	anim12d 608	. . . . . . . 8 ⊢ (𝐻:𝐴⟶𝐵 → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵)))
9	2, 3, 8	3syl 18	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵)))
10	9	imp 407	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵))
11		breq1 4971	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑎𝑆𝑏 ↔ (𝐻‘𝑐)𝑆𝑏))
12		eqeq1 2801	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑎 = 𝑏 ↔ (𝐻‘𝑐) = 𝑏))
13		breq2 4972	. . . . . . . 8 ⊢ (𝑎 = (𝐻‘𝑐) → (𝑏𝑆𝑎 ↔ 𝑏𝑆(𝐻‘𝑐)))
14	11, 12, 13	3orbi123d 1427	. . . . . . 7 ⊢ (𝑎 = (𝐻‘𝑐) → ((𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) ↔ ((𝐻‘𝑐)𝑆𝑏 ∨ (𝐻‘𝑐) = 𝑏 ∨ 𝑏𝑆(𝐻‘𝑐))))
15		breq2 4972	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → ((𝐻‘𝑐)𝑆𝑏 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑑)))
16		eqeq2 2808	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → ((𝐻‘𝑐) = 𝑏 ↔ (𝐻‘𝑐) = (𝐻‘𝑑)))
17		breq1 4971	. . . . . . . 8 ⊢ (𝑏 = (𝐻‘𝑑) → (𝑏𝑆(𝐻‘𝑐) ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
18	15, 16, 17	3orbi123d 1427	. . . . . . 7 ⊢ (𝑏 = (𝐻‘𝑑) → (((𝐻‘𝑐)𝑆𝑏 ∨ (𝐻‘𝑐) = 𝑏 ∨ 𝑏𝑆(𝐻‘𝑐)) ↔ ((𝐻‘𝑐)𝑆(𝐻‘𝑑) ∨ (𝐻‘𝑐) = (𝐻‘𝑑) ∨ (𝐻‘𝑑)𝑆(𝐻‘𝑐))))
19	14, 18	rspc2v 3574	. . . . . 6 ⊢ (((𝐻‘𝑐) ∈ 𝐵 ∧ (𝐻‘𝑑) ∈ 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) → ((𝐻‘𝑐)𝑆(𝐻‘𝑑) ∨ (𝐻‘𝑐) = (𝐻‘𝑑) ∨ (𝐻‘𝑑)𝑆(𝐻‘𝑐))))
20	10, 19	syl 17	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) → ((𝐻‘𝑐)𝑆(𝐻‘𝑑) ∨ (𝐻‘𝑐) = (𝐻‘𝑑) ∨ (𝐻‘𝑑)𝑆(𝐻‘𝑐))))
21		isorel 6949	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑐𝑅𝑑 ↔ (𝐻‘𝑐)𝑆(𝐻‘𝑑)))
22		f1of1 6489	. . . . . . . . 9 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
23	2, 22	syl 17	. . . . . . . 8 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1→𝐵)
24		f1fveq 6892	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) = (𝐻‘𝑑) ↔ 𝑐 = 𝑑))
25	23, 24	sylan 580	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝐻‘𝑐) = (𝐻‘𝑑) ↔ 𝑐 = 𝑑))
26	25	bicomd 224	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑐 = 𝑑 ↔ (𝐻‘𝑐) = (𝐻‘𝑑)))
27		isorel 6949	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
28	27	ancom2s 646	. . . . . 6 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑑𝑅𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑐)))
29	21, 26, 28	3orbi123d 1427	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → ((𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐) ↔ ((𝐻‘𝑐)𝑆(𝐻‘𝑑) ∨ (𝐻‘𝑐) = (𝐻‘𝑑) ∨ (𝐻‘𝑑)𝑆(𝐻‘𝑐))))
30	20, 29	sylibrd 260	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) → (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
31	30	ralrimdvva 3163	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎) → ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
32	1, 31	anim12d 608	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎)) → (𝑅 Po 𝐴 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐))))
33		df-so 5370	. 2 ⊢ (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎𝑆𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑆𝑎)))
34		df-so 5370	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑐 ∈ 𝐴 ∀𝑑 ∈ 𝐴 (𝑐𝑅𝑑 ∨ 𝑐 = 𝑑 ∨ 𝑑𝑅𝑐)))
35	32, 33, 34	3imtr4g 297	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ w3o 1079   = wceq 1525   ∈ wcel 2083  ∀wral 3107   class class class wbr 4968   Po wpo 5367   Or wor 5368  ⟶wf 6228  –1-1→wf1 6229  –1-1-onto→wf1o 6231  ‘cfv 6232   Isom wiso 6233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-po 5369  df-so 5370  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-f1o 6239  df-fv 6240  df-isom 6241

This theorem is referenced by:  isoso  6971  isowe2  6973