Theorem isopolem 5865

Description: Lemma for isopo 5866. (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 5850	. . . . . . . . . . 11 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
2		f1of 5500	. . . . . . . . . . 11 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
3		ffvelcdm 5691	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑑 ∈ 𝐴) → (𝐻‘𝑑) ∈ 𝐵)
4	3	ex 115	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑑 ∈ 𝐴 → (𝐻‘𝑑) ∈ 𝐵))
5		ffvelcdm 5691	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑒 ∈ 𝐴) → (𝐻‘𝑒) ∈ 𝐵)
6	5	ex 115	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑒 ∈ 𝐴 → (𝐻‘𝑒) ∈ 𝐵))
7		ffvelcdm 5691	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑓 ∈ 𝐴) → (𝐻‘𝑓) ∈ 𝐵)
8	7	ex 115	. . . . . . . . . . . 12 ⊢ (𝐻:𝐴⟶𝐵 → (𝑓 ∈ 𝐴 → (𝐻‘𝑓) ∈ 𝐵))
9	4, 6, 8	3anim123d 1330	. . . . . . . . . . 11 ⊢ (𝐻:𝐴⟶𝐵 → ((𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴) → ((𝐻‘𝑑) ∈ 𝐵 ∧ (𝐻‘𝑒) ∈ 𝐵 ∧ (𝐻‘𝑓) ∈ 𝐵)))
10	1, 2, 9	3syl 17	. . . . . . . . . 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 124	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → ((𝐻‘𝑑) ∈ 𝐵 ∧ (𝐻‘𝑒) ∈ 𝐵 ∧ (𝐻‘𝑓) ∈ 𝐵))
12		breq12 4034	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝐻‘𝑑) ∧ 𝑎 = (𝐻‘𝑑)) → (𝑎𝑆𝑎 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
13	12	anidms 397	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐻‘𝑑) → (𝑎𝑆𝑎 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
14	13	notbid 668	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐻‘𝑑) → (¬ 𝑎𝑆𝑎 ↔ ¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
15		breq1 4032	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐻‘𝑑) → (𝑎𝑆𝑏 ↔ (𝐻‘𝑑)𝑆𝑏))
16	15	anbi1d 465	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐻‘𝑑) → ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) ↔ ((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐)))
17		breq1 4032	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐻‘𝑑) → (𝑎𝑆𝑐 ↔ (𝐻‘𝑑)𝑆𝑐))
18	16, 17	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐻‘𝑑) → (((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐) ↔ (((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐)))
19	14, 18	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑎 = (𝐻‘𝑑) → ((¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)) ↔ (¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑) ∧ (((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐) → (𝐻‘𝑑)𝑆𝑐))))
20		breq2 4033	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝐻‘𝑒) → ((𝐻‘𝑑)𝑆𝑏 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑒)))
21		breq1 4032	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝐻‘𝑒) → (𝑏𝑆𝑐 ↔ (𝐻‘𝑒)𝑆𝑐))
22	20, 21	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝐻‘𝑒) → (((𝐻‘𝑑)𝑆𝑏 ∧ 𝑏𝑆𝑐) ↔ ((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆𝑐)))
23	22	imbi1d 231	. . . . . . . . . . 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 464	. . . . . . . . . 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 4033	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝐻‘𝑓) → ((𝐻‘𝑒)𝑆𝑐 ↔ (𝐻‘𝑒)𝑆(𝐻‘𝑓)))
26	25	anbi2d 464	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝐻‘𝑓) → (((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆𝑐) ↔ ((𝐻‘𝑑)𝑆(𝐻‘𝑒) ∧ (𝐻‘𝑒)𝑆(𝐻‘𝑓))))
27		breq2 4033	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝐻‘𝑓) → ((𝐻‘𝑑)𝑆𝑐 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑓)))
28	26, 27	imbi12d 234	. . . . . . . . . . 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 464	. . . . . . . . . 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 2880	. . . . . . . . 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 14	. . . . . . . 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 109	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
33		simpr1 1005	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → 𝑑 ∈ 𝐴)
34		isorel 5851	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
35	32, 33, 33, 34	syl12anc 1247	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑑𝑅𝑑 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
36	35	notbid 668	. . . . . . . . 9 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (¬ 𝑑𝑅𝑑 ↔ ¬ (𝐻‘𝑑)𝑆(𝐻‘𝑑)))
37		simpr2 1006	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → 𝑒 ∈ 𝐴)
38		isorel 5851	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑒)))
39	32, 33, 37, 38	syl12anc 1247	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑑𝑅𝑒 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑒)))
40		simpr3 1007	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → 𝑓 ∈ 𝐴)
41		isorel 5851	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻‘𝑒)𝑆(𝐻‘𝑓)))
42	32, 37, 40, 41	syl12anc 1247	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑒𝑅𝑓 ↔ (𝐻‘𝑒)𝑆(𝐻‘𝑓)))
43	39, 42	anbi12d 473	. . . . . . . . . 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 5851	. . . . . . . . . . 11 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑓)))
45	32, 33, 40, 44	syl12anc 1247	. . . . . . . . . 10 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ∧ 𝑓 ∈ 𝐴)) → (𝑑𝑅𝑓 ↔ (𝐻‘𝑑)𝑆(𝐻‘𝑓)))
46	43, 45	imbi12d 234	. . . . . . . . 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 473	. . . . . . . 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 169	. . . . . . 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 115	. . . . . 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 78	. . . . 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 256	. . . 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 2577	. . 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 115	. 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 4327	. 2 ⊢ (𝑆 Po 𝐵 ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (¬ 𝑎𝑆𝑎 ∧ ((𝑎𝑆𝑏 ∧ 𝑏𝑆𝑐) → 𝑎𝑆𝑐)))
55		df-po 4327	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐴 ∀𝑓 ∈ 𝐴 (¬ 𝑑𝑅𝑑 ∧ ((𝑑𝑅𝑒 ∧ 𝑒𝑅𝑓) → 𝑑𝑅𝑓)))
56	53, 54, 55	3imtr4g 205	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472   class class class wbr 4029   Po wpo 4325  ⟶wf 5250  –1-1-onto→wf1o 5253  ‘cfv 5254   Isom wiso 5255

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-f1o 5261  df-fv 5262  df-isom 5263

This theorem is referenced by:  isopo  5866  isosolem  5867