Theorem isopolem 7344

Description: Lemma for isopo 7345. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Assertion

Ref	Expression
isopolem	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))

Proof of Theorem isopolem

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

Step	Hyp	Ref	Expression
1		isof1o 7322	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
2		f1of 6821	. . . . . . . . . . 11 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
3		ffvelcdm 7077	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑑 ∈ 𝐴) → (𝐻‘𝑑) ∈ 𝐵)
4	3	ex 417	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑑 ∈ 𝐴 → (𝐻‘𝑑) ∈ 𝐵))
5		ffvelcdm 7077	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑒 ∈ 𝐴) → (𝐻‘𝑒) ∈ 𝐵)
6	5	ex 417	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑒 ∈ 𝐴 → (𝐻‘𝑒) ∈ 𝐵))
7		ffvelcdm 7077	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑓 ∈ 𝐴) → (𝐻‘𝑓) ∈ 𝐵)
8	7	ex 417	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑓 ∈ 𝐴 → (𝐻‘𝑓) ∈ 𝐵))
9	4, 6, 8	3anim123d 1469	. . . . . . . . . . 11 ⊢ (𝐻:𝐴⟶𝐵 → ((𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴) → ((𝐻‘𝑑) ∈ 𝐵 ∧ (𝐻‘𝑒) ∈ 𝐵 ∧ (𝐻‘𝑓) ∈ 𝐵)))
10	1, 2, 9	3syl 19	. . . . . . . . . 10 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴) → ((𝐻‘𝑑) ∈ 𝐵 ∧ (𝐻‘𝑒) ∈ 𝐵 ∧ (𝐻‘𝑓) ∈ 𝐵)))
11	10	imp 411	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → ((𝐻‘𝑑) ∈ 𝐵 ∧ (𝐻‘𝑒) ∈ 𝐵 ∧ (𝐻‘𝑓) ∈ 𝐵))
12		breq12 5118	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝐻‘𝑑) ∧ 𝑎 = (𝐻‘𝑑)) → (𝑎𝑆𝑎 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
13	12	anidms 576	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐻‘𝑑) → (𝑎𝑆𝑎 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
14	13	notbid 321	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐻‘𝑑) → (¬ 𝑎𝑆𝑎 ↔ ¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
15		breq1 5116	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐻‘𝑑) → (𝑎𝑆𝑏 ↔ (𝐻‘𝑑)𝑆𝑏))
16	15	anbi1d 642	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐻‘𝑑) → ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) ↔ ((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐)))
17		breq1 5116	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐻‘𝑑) → (𝑎𝑆𝑐 ↔ (𝐻‘𝑑)𝑆𝑐))
18	16, 17	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐻‘𝑑) → (((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐) ↔ (((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐)))
19	14, 18	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑎 = (𝐻‘𝑑) → ((¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)) ↔ (¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑) ∧ (((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐))))
20		breq2 5117	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝐻‘𝑒) → ((𝐻‘𝑑)𝑆𝑏 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑒)))
21		breq1 5116	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝐻‘𝑒) → (𝑏𝑆𝑐 ↔ (𝐻‘𝑒)𝑆𝑐))
22	20, 21	anbi12d 643	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝐻‘𝑒) → (((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐) ↔ ((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆𝑐)))
23	22	imbi1d 344	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐻‘𝑒) → ((((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐) ↔ (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐)))
24	23	anbi2d 641	. . . . . . . . . 10 ⊢ (𝑏 = (𝐻‘𝑒) → ((¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑) ∧ (((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐)) ↔ (¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑) ∧ (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐))))
25		breq2 5117	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝐻‘𝑓) → ((𝐻‘𝑒)𝑆𝑐 ↔ (𝐻‘𝑒)𝑆(𝐻‘𝑓)))
26	25	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝐻‘𝑓) → (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆𝑐) ↔ ((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆(𝐻‘𝑓))))
27		breq2 5117	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝐻‘𝑓) → ((𝐻‘𝑑)𝑆𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑓)))
28	26, 27	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑐 = (𝐻‘𝑓) → ((((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐) ↔ (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆(𝐻‘𝑓)) → (𝐻‘𝑑)𝑆(𝐻‘𝑓))))
29	28	anbi2d 641	. . . . . . . . . 10 ⊢ (𝑐 = (𝐻‘𝑓) → ((¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑) ∧ (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐)) ↔ (¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑) ∧ (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆(𝐻‘𝑓)) → (𝐻‘𝑑)𝑆(𝐻‘𝑓)))))
30	19, 24, 29	rspc3v 3606	. . . . . . . . 9 ⊢ (((𝐻‘𝑑) ∈ 𝐵 ∧ (𝐻‘𝑒) ∈ 𝐵 ∧ (𝐻‘𝑓) ∈ 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑) ∧ (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆(𝐻‘𝑓)) → (𝐻‘𝑑)𝑆(𝐻‘𝑓)))))
31	11, 30	syl 18	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑) ∧ (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆(𝐻‘𝑓)) → (𝐻‘𝑑)𝑆(𝐻‘𝑓)))))
32		simpl 487	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
33		simpr1 1211	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → 𝑑 ∈ 𝐴)
34		isorel 7325	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
35	32, 33, 33, 34	syl12anc 849	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
36	35	notbid 321	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (¬ 𝑑𝑅𝑑 ↔ ¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
37		simpr2 1212	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → 𝑒 ∈ 𝐴)
38		isorel 7325	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑒)))
39	32, 33, 37, 38	syl12anc 849	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑒)))
40		simpr3 1213	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → 𝑓 ∈ 𝐴)
41		isorel 7325	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻‘𝑒)𝑆(𝐻‘𝑓)))
42	32, 37, 40, 41	syl12anc 849	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻‘𝑒)𝑆(𝐻‘𝑓)))
43	39, 42	anbi12d 643	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) ↔ ((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆(𝐻‘𝑓))))
44		isorel 7325	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑓)))
45	32, 33, 40, 44	syl12anc 849	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑓)))
46	43, 45	imbi12d 347	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓) ↔ (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆(𝐻‘𝑓)) → (𝐻‘𝑑)𝑆(𝐻‘𝑓))))
47	36, 46	anbi12d 643	. . . . . . . 8 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → ((¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓)) ↔ (¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑) ∧ (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆(𝐻‘𝑓)) → (𝐻‘𝑑)𝑆(𝐻‘𝑓)))))
48	31, 47	sylibrd 262	. . . . . . 7 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓))))
49	48	ex 417	. . . . . 6 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓)))))
50	49	com23 87	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → ((𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓)))))
51	50	imp31 422	. . . 4 ⊢ (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐))) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
52	51	ralrimivvva 3217	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐))) → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐴 ∀𝑓 ∈ 𝐴 (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
53	52	ex 417	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)) → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐴 ∀𝑓 ∈ 𝐴 (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓))))
54		df-po 5570	. 2 ⊢ (𝑆 Po 𝐵 ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)))
55		df-po 5570	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐴 ∀𝑓 ∈ 𝐴 (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
56	53, 54, 55	3imtr4g 299	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   class class class wbr 5113   Po wpo 5568  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537   Isom wiso 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-po 5570  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-f1o 6544  df-fv 6545  df-isom 6546

This theorem is referenced by:  isopo  7345  isosolem  7346