Theorem ofoafo 44111

Description: Addition operator for functions from a set into a power of omega is an onto binary operator. (Contributed by RP, 5-Jan-2025.)

Assertion

Ref	Expression
ofoafo	⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))):((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))–onto→(𝐶 ↑m 𝐴))

Proof of Theorem ofoafo

Dummy variables 𝑎 𝑓 ℎ 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
2		inidm 4178	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	eqcomi 2771	. . . . 5 ⊢ 𝐴 = (𝐴 ∩ 𝐴)
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = (𝐴 ∩ 𝐴))
5	1, 1, 4	3jca 1145	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 = (𝐴 ∩ 𝐴)))
6		ofoaf 44110	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 = (𝐴 ∩ 𝐴)) ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))):((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))⟶(𝐶 ↑m 𝐴))
7	5, 6	sylan 591	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))):((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))⟶(𝐶 ↑m 𝐴))
8		simpr 489	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → ℎ ∈ (𝐶 ↑m 𝐴))
9		omelon 9613	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
10	9	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ω ∈ On)
11		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐵 ∈ On)
12	10, 11	jca 520	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ∈ On ∧ 𝐵 ∈ On))
13		peano1 7883	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
14		oen0 8570	. . . . . . . . . . . . 13 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
15	12, 13, 14	sylancl 597	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ (ω ↑o 𝐵))
16		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 = (ω ↑o 𝐵))
17	15, 16	eleqtrrd 2865	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ 𝐶)
18		fconst6g 6767	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝐶 → (𝐴 × {∅}):𝐴⟶𝐶)
19	17, 18	syl 18	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (𝐴 × {∅}):𝐴⟶𝐶)
20	19	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}):𝐴⟶𝐶)
21		oecl 8520	. . . . . . . . . . . . 13 ⊢ ((ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑o 𝐵) ∈ On)
22	9, 11, 21	sylancr 598	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ↑o 𝐵) ∈ On)
23	16, 22	eqeltrd 2862	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 ∈ On)
24	23	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐶 ∈ On)
25		simpl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐴 ∈ 𝑉)
26	24, 25	elmapd 8835	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) ↔ (𝐴 × {∅}):𝐴⟶𝐶))
27	20, 26	mpbird 260	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))
28	27	adantr 485	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))
29		ovres 7578	. . . . . . . . . 10 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))(𝐴 × {∅})) = (ℎ ∘f +o (𝐴 × {∅})))
30	29	adantl 486	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))(𝐴 × {∅})) = (ℎ ∘f +o (𝐴 × {∅})))
31		elmapi 8844	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝐶 ↑m 𝐴) → ℎ:𝐴⟶𝐶)
32	31	adantr 485	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → ℎ:𝐴⟶𝐶)
33	32	ffnd 6706	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → ℎ Fn 𝐴)
34	33	adantl 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → ℎ Fn 𝐴)
35		elmapi 8844	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) → (𝐴 × {∅}):𝐴⟶𝐶)
36	35	adantl 486	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (𝐴 × {∅}):𝐴⟶𝐶)
37	36	ffnd 6706	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
38	37	adantl 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → (𝐴 × {∅}) Fn 𝐴)
39	25	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → 𝐴 ∈ 𝑉)
40	34, 38, 39, 39, 2	offn 7689	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → (ℎ ∘f +o (𝐴 × {∅})) Fn 𝐴)
41		elmapfn 8860	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝐶 ↑m 𝐴) → ℎ Fn 𝐴)
42		elmapfn 8860	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) → (𝐴 × {∅}) Fn 𝐴)
43	41, 42	anim12i 624	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (ℎ Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
44	43	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → (ℎ Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
45	39	anim1i 626	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴))
46		fnfvof 7693	. . . . . . . . . . . 12 ⊢ (((ℎ Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴)) → ((ℎ ∘f +o (𝐴 × {∅}))‘𝑎) = ((ℎ‘𝑎) +o ((𝐴 × {∅})‘𝑎)))
47	44, 45, 46	syl2an2r 697	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((ℎ ∘f +o (𝐴 × {∅}))‘𝑎) = ((ℎ‘𝑎) +o ((𝐴 × {∅})‘𝑎)))
48		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
49		fvconst2g 7200	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ ω ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
50	13, 48, 49	sylancr 598	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
51	50	oveq2d 7428	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((ℎ‘𝑎) +o ((𝐴 × {∅})‘𝑎)) = ((ℎ‘𝑎) +o ∅))
52	24	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → 𝐶 ∈ On)
53	52	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ On)
54		onss 7782	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
55	53, 54	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ On)
56	31	ad2antrl 740	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → ℎ:𝐴⟶𝐶)
57	56	ffvelcdmda 7079	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (ℎ‘𝑎) ∈ 𝐶)
58	55, 57	sseldd 3937	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (ℎ‘𝑎) ∈ On)
59		oa0 8499	. . . . . . . . . . . 12 ⊢ ((ℎ‘𝑎) ∈ On → ((ℎ‘𝑎) +o ∅) = (ℎ‘𝑎))
60	58, 59	syl 18	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((ℎ‘𝑎) +o ∅) = (ℎ‘𝑎))
61	47, 51, 60	3eqtrd 2801	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((ℎ ∘f +o (𝐴 × {∅}))‘𝑎) = (ℎ‘𝑎))
62	40, 34, 61	eqfnfvd 7028	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → (ℎ ∘f +o (𝐴 × {∅})) = ℎ)
63	30, 62	eqtr2d 2798	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))(𝐴 × {∅})))
64	63	expr 461	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) → ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))(𝐴 × {∅}))))
65	28, 64	jcai 525	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) ∧ ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))(𝐴 × {∅}))))
66		oveq2 7420	. . . . . . 7 ⊢ (𝑧 = (𝐴 × {∅}) → (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧) = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))(𝐴 × {∅})))
67	66	rspceeqv 3603	. . . . . 6 ⊢ (((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) ∧ ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))(𝐴 × {∅}))) → ∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧))
68	65, 67	syl 18	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → ∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧))
69	8, 68	jca 520	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → (ℎ ∈ (𝐶 ↑m 𝐴) ∧ ∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧)))
70		oveq1 7419	. . . . . . 7 ⊢ (𝑓 = ℎ → (𝑓( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧) = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧))
71	70	eqeq2d 2773	. . . . . 6 ⊢ (𝑓 = ℎ → (ℎ = (𝑓( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧) ↔ ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧)))
72	71	rexbidv 3188	. . . . 5 ⊢ (𝑓 = ℎ → (∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (𝑓( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧) ↔ ∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧)))
73	72	rspcev 3580	. . . 4 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ ∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (ℎ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧)) → ∃𝑓 ∈ (𝐶 ↑m 𝐴)∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (𝑓( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧))
74	69, 73	syl 18	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → ∃𝑓 ∈ (𝐶 ↑m 𝐴)∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (𝑓( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧))
75	74	ralrimiva 3156	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ∀ℎ ∈ (𝐶 ↑m 𝐴)∃𝑓 ∈ (𝐶 ↑m 𝐴)∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (𝑓( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧))
76		foov 7586	. 2 ⊢ (( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))):((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))–onto→(𝐶 ↑m 𝐴) ↔ (( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))):((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))⟶(𝐶 ↑m 𝐴) ∧ ∀ℎ ∈ (𝐶 ↑m 𝐴)∃𝑓 ∈ (𝐶 ↑m 𝐴)∃𝑧 ∈ (𝐶 ↑m 𝐴)ℎ = (𝑓( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴)))𝑧)))
77	7, 75, 76	sylanbrc 594	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))):((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))–onto→(𝐶 ↑m 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4588   × cxp 5658   ↾ cres 5662  Oncon0 6360   Fn wfn 6531  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7412   ∘_f cof 7674  ωcom 7860   +_o coa 8448   ↑_o coe 8450   ↑_m cmap 8822

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-inf2 9608

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-omul 8456  df-oexp 8457  df-map 8824

This theorem is used by: (None)