Theorem ofoaass 43788

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 8812	. . . . . 6 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹 Fn 𝐴)
2	1	3ad2ant1 1134	. . . . 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 8812	. . . . . 6 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐴) → 𝐺 Fn 𝐴)
5	4	3ad2ant2 1135	. . . . 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 767	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐴 ∈ 𝑉)
8		inidm 4167	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9	3, 6, 7, 7, 8	offn 7644	. . 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 8812	. . . . 5 ⊢ (𝐻 ∈ (𝐵 ↑m 𝐴) → 𝐻 Fn 𝐴)
11	10	3ad2ant3 1136	. . . 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 7644	. 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 7644	. . 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 7644	. 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 776	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ On)
17		elmapi 8796	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹:𝐴⟶𝐵)
18	17	3ad2ant1 1134	. . . . . . . 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 7036	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝐵)
21		onelon 6348	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ (𝐹‘𝑎) ∈ 𝐵) → (𝐹‘𝑎) ∈ On)
22	16, 20, 21	syl2anc 585	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ On)
23		elmapi 8796	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐴) → 𝐺:𝐴⟶𝐵)
24	23	3ad2ant2 1135	. . . . . . . 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 7036	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ 𝐵)
27		onelon 6348	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ (𝐺‘𝑎) ∈ 𝐵) → (𝐺‘𝑎) ∈ On)
28	16, 26, 27	syl2anc 585	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ On)
29		elmapi 8796	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝐵 ↑m 𝐴) → 𝐻:𝐴⟶𝐵)
30	29	3ad2ant3 1136	. . . . . . . 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 7036	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ 𝐵)
33		onelon 6348	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ (𝐻‘𝑎) ∈ 𝐵) → (𝐻‘𝑎) ∈ On)
34	16, 32, 33	syl2anc 585	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ On)
35		oaass 8496	. . . . 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 1374	. . . 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 616	. . . . . 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 7648	. . . . . 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 838	. . . . 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 7382	. . . 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 7648	. . . . . 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 686	. . . . 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 7383	. . . 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 2781	. . 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 7648	. . . 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 686	. . 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 7648	. . . 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 686	. . 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 2781	. 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 6986	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 1087   = wceq 1542   ∈ wcel 2114  Oncon0 6323   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629   +_o coa 8402   ↑_m cmap 8773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409  df-map 8775

This theorem is referenced by: (None)