Theorem ofoaass 42668

Description: Component-wise addition of ordinal-yielding functions is associative. (Contributed by RP, 5-Jan-2025.)

Assertion

Ref	Expression
ofoaass	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))

Proof of Theorem ofoaass

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapfn 8858	. . . . . 6 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹 Fn 𝐴)
2	1	3ad2ant1 1130	. . . . 5 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐹 Fn 𝐴)
3	2	adantl 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐹 Fn 𝐴)
4		elmapfn 8858	. . . . . 6 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐴) → 𝐺 Fn 𝐴)
5	4	3ad2ant2 1131	. . . . 5 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐺 Fn 𝐴)
6	5	adantl 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐺 Fn 𝐴)
7		simpll 764	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐴 ∈ 𝑉)
8		inidm 4213	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9	3, 6, 7, 7, 8	offn 7679	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) Fn 𝐴)
10		elmapfn 8858	. . . . 5 ⊢ (𝐻 ∈ (𝐵 ↑m 𝐴) → 𝐻 Fn 𝐴)
11	10	3ad2ant3 1132	. . . 4 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐻 Fn 𝐴)
12	11	adantl 481	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐻 Fn 𝐴)
13	9, 12, 7, 7, 8	offn 7679	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) Fn 𝐴)
14	6, 12, 7, 7, 8	offn 7679	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → (𝐺 ∘f +o 𝐻) Fn 𝐴)
15	3, 14, 7, 7, 8	offn 7679	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)) Fn 𝐴)
16		simpllr 773	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ On)
17		elmapi 8842	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹:𝐴⟶𝐵)
18	17	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐹:𝐴⟶𝐵)
19	18	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐹:𝐴⟶𝐵)
20	19	ffvelcdmda 7079	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝐵)
21		onelon 6382	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ (𝐹‘𝑎) ∈ 𝐵) → (𝐹‘𝑎) ∈ On)
22	16, 20, 21	syl2anc 583	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ On)
23		elmapi 8842	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐴) → 𝐺:𝐴⟶𝐵)
24	23	3ad2ant2 1131	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐺:𝐴⟶𝐵)
25	24	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐺:𝐴⟶𝐵)
26	25	ffvelcdmda 7079	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ 𝐵)
27		onelon 6382	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ (𝐺‘𝑎) ∈ 𝐵) → (𝐺‘𝑎) ∈ On)
28	16, 26, 27	syl2anc 583	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ On)
29		elmapi 8842	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝐵 ↑m 𝐴) → 𝐻:𝐴⟶𝐵)
30	29	3ad2ant3 1132	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐻:𝐴⟶𝐵)
31	30	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐻:𝐴⟶𝐵)
32	31	ffvelcdmda 7079	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ 𝐵)
33		onelon 6382	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ (𝐻‘𝑎) ∈ 𝐵) → (𝐻‘𝑎) ∈ On)
34	16, 32, 33	syl2anc 583	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ On)
35		oaass 8559	. . . . 5 ⊢ (((𝐹‘𝑎) ∈ On ∧ (𝐺‘𝑎) ∈ On ∧ (𝐻‘𝑎) ∈ On) → (((𝐹‘𝑎) +o (𝐺‘𝑎)) +o (𝐻‘𝑎)) = ((𝐹‘𝑎) +o ((𝐺‘𝑎) +o (𝐻‘𝑎))))
36	22, 28, 34, 35	syl3anc 1368	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (((𝐹‘𝑎) +o (𝐺‘𝑎)) +o (𝐻‘𝑎)) = ((𝐹‘𝑎) +o ((𝐺‘𝑎) +o (𝐻‘𝑎))))
37	3	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐹 Fn 𝐴)
38	6	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐺 Fn 𝐴)
39	7	anim1i 614	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴))
40		fnfvof 7683	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴)) → ((𝐹 ∘f +o 𝐺)‘𝑎) = ((𝐹‘𝑎) +o (𝐺‘𝑎)))
41	37, 38, 39, 40	syl21anc 835	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐹 ∘f +o 𝐺)‘𝑎) = ((𝐹‘𝑎) +o (𝐺‘𝑎)))
42	41	oveq1d 7419	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (((𝐹 ∘f +o 𝐺)‘𝑎) +o (𝐻‘𝑎)) = (((𝐹‘𝑎) +o (𝐺‘𝑎)) +o (𝐻‘𝑎)))
43	6, 12	jca 511	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → (𝐺 Fn 𝐴 ∧ 𝐻 Fn 𝐴))
44		fnfvof 7683	. . . . . 6 ⊢ (((𝐺 Fn 𝐴 ∧ 𝐻 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴)) → ((𝐺 ∘f +o 𝐻)‘𝑎) = ((𝐺‘𝑎) +o (𝐻‘𝑎)))
45	43, 39, 44	syl2an2r 682	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐺 ∘f +o 𝐻)‘𝑎) = ((𝐺‘𝑎) +o (𝐻‘𝑎)))
46	45	oveq2d 7420	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) +o ((𝐺 ∘f +o 𝐻)‘𝑎)) = ((𝐹‘𝑎) +o ((𝐺‘𝑎) +o (𝐻‘𝑎))))
47	36, 42, 46	3eqtr4d 2776	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (((𝐹 ∘f +o 𝐺)‘𝑎) +o (𝐻‘𝑎)) = ((𝐹‘𝑎) +o ((𝐺 ∘f +o 𝐻)‘𝑎)))
48	9, 12	jca 511	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → ((𝐹 ∘f +o 𝐺) Fn 𝐴 ∧ 𝐻 Fn 𝐴))
49		fnfvof 7683	. . . 4 ⊢ ((((𝐹 ∘f +o 𝐺) Fn 𝐴 ∧ 𝐻 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴)) → (((𝐹 ∘f +o 𝐺) ∘f +o 𝐻)‘𝑎) = (((𝐹 ∘f +o 𝐺)‘𝑎) +o (𝐻‘𝑎)))
50	48, 39, 49	syl2an2r 682	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (((𝐹 ∘f +o 𝐺) ∘f +o 𝐻)‘𝑎) = (((𝐹 ∘f +o 𝐺)‘𝑎) +o (𝐻‘𝑎)))
51	3, 14	jca 511	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → (𝐹 Fn 𝐴 ∧ (𝐺 ∘f +o 𝐻) Fn 𝐴))
52		fnfvof 7683	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ (𝐺 ∘f +o 𝐻) Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴)) → ((𝐹 ∘f +o (𝐺 ∘f +o 𝐻))‘𝑎) = ((𝐹‘𝑎) +o ((𝐺 ∘f +o 𝐻)‘𝑎)))
53	51, 39, 52	syl2an2r 682	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐹 ∘f +o (𝐺 ∘f +o 𝐻))‘𝑎) = ((𝐹‘𝑎) +o ((𝐺 ∘f +o 𝐻)‘𝑎)))
54	47, 50, 53	3eqtr4d 2776	. 2 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (((𝐹 ∘f +o 𝐺) ∘f +o 𝐻)‘𝑎) = ((𝐹 ∘f +o (𝐺 ∘f +o 𝐻))‘𝑎))
55	13, 15, 54	eqfnfvd 7028	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Oncon0 6357   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404   ∘_f cof 7664   +_o coa 8461   ↑_m cmap 8819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-oadd 8468  df-map 8821

This theorem is referenced by: (None)