Theorem ofoaass 41277

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 8702	. . . . . 6 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹 Fn 𝐴)
2	1	3ad2ant1 1132	. . . . 5 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐹 Fn 𝐴)
3	2	adantl 482	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐹 Fn 𝐴)
4		elmapfn 8702	. . . . . 6 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐴) → 𝐺 Fn 𝐴)
5	4	3ad2ant2 1133	. . . . 5 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐺 Fn 𝐴)
6	5	adantl 482	. . . 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 4162	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9	3, 6, 7, 7, 8	offn 7587	. . 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 8702	. . . . 5 ⊢ (𝐻 ∈ (𝐵 ↑m 𝐴) → 𝐻 Fn 𝐴)
11	10	3ad2ant3 1134	. . . 4 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐻 Fn 𝐴)
12	11	adantl 482	. . 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 7587	. 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 7587	. . 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 7587	. 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 8686	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹:𝐴⟶𝐵)
18	17	3ad2ant1 1132	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐹:𝐴⟶𝐵)
19	18	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐹:𝐴⟶𝐵)
20	19	ffvelcdmda 7000	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝐵)
21		onelon 6313	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ (𝐹‘𝑎) ∈ 𝐵) → (𝐹‘𝑎) ∈ On)
22	16, 20, 21	syl2anc 584	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ On)
23		elmapi 8686	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐴) → 𝐺:𝐴⟶𝐵)
24	23	3ad2ant2 1133	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐺:𝐴⟶𝐵)
25	24	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐺:𝐴⟶𝐵)
26	25	ffvelcdmda 7000	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ 𝐵)
27		onelon 6313	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ (𝐺‘𝑎) ∈ 𝐵) → (𝐺‘𝑎) ∈ On)
28	16, 26, 27	syl2anc 584	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ On)
29		elmapi 8686	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝐵 ↑m 𝐴) → 𝐻:𝐴⟶𝐵)
30	29	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴)) → 𝐻:𝐴⟶𝐵)
31	30	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐻:𝐴⟶𝐵)
32	31	ffvelcdmda 7000	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ 𝐵)
33		onelon 6313	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ (𝐻‘𝑎) ∈ 𝐵) → (𝐻‘𝑎) ∈ On)
34	16, 32, 33	syl2anc 584	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ On)
35		oaass 8441	. . . . 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 1370	. . . 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 481	. . . . . 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 481	. . . . . 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 615	. . . . . 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 7591	. . . . . 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 7331	. . . 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 512	. . . . . 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 7591	. . . . . 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 7332	. . . 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 2786	. . 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 512	. . . 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 7591	. . . 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 512	. . . 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 7591	. . . 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 2786	. 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 6951	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 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  Oncon0 6288   Fn wfn 6460  ⟶wf 6461  ‘cfv 6465  (class class class)co 7316   ∘_f cof 7572   +_o coa 8342   ↑_m cmap 8664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-int 4892  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-ov 7319  df-oprab 7320  df-mpo 7321  df-of 7574  df-om 7759  df-1st 7877  df-2nd 7878  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-oadd 8349  df-map 8666

This theorem is referenced by: (None)