Theorem ofoaass 43452

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 8789	. . . . . 6 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹 Fn 𝐴)
2	1	3ad2ant1 1133	. . . . 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 8789	. . . . . 6 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐴) → 𝐺 Fn 𝐴)
5	4	3ad2ant2 1134	. . . . 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 766	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → 𝐴 ∈ 𝑉)
8		inidm 4174	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9	3, 6, 7, 7, 8	offn 7623	. . 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 8789	. . . . 5 ⊢ (𝐻 ∈ (𝐵 ↑m 𝐴) → 𝐻 Fn 𝐴)
11	10	3ad2ant3 1135	. . . 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 7623	. 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 7623	. . 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 7623	. 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 775	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ On)
17		elmapi 8773	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) → 𝐹:𝐴⟶𝐵)
18	17	3ad2ant1 1133	. . . . . . . 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 7017	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝐵)
21		onelon 6331	. . . . . 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 8773	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐵 ↑m 𝐴) → 𝐺:𝐴⟶𝐵)
24	23	3ad2ant2 1134	. . . . . . . 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 7017	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ 𝐵)
27		onelon 6331	. . . . . 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 8773	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝐵 ↑m 𝐴) → 𝐻:𝐴⟶𝐵)
30	29	3ad2ant3 1135	. . . . . . . 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 7017	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐻‘𝑎) ∈ 𝐵)
33		onelon 6331	. . . . . 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 8476	. . . . 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 1373	. . . 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 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 7627	. . . . . 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 837	. . . . 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 7361	. . . 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 7627	. . . . . 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 685	. . . . 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 7362	. . . 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 7627	. . . 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 685	. . 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 7627	. . . 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 685	. . 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 6967	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 1086   = wceq 1541   ∈ wcel 2111  Oncon0 6306   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∘_f cof 7608   +_o coa 8382   ↑_m cmap 8750

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-oadd 8389  df-map 8752

This theorem is referenced by: (None)