Theorem ofoafo 43339

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 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
2		inidm 4178	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	eqcomi 2738	. . . . 5 ⊢ 𝐴 = (𝐴 ∩ 𝐴)
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = (𝐴 ∩ 𝐴))
5	1, 1, 4	3jca 1128	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 = (𝐴 ∩ 𝐴)))
6		ofoaf 43338	. . 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 580	. 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 484	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → ℎ ∈ (𝐶 ↑m 𝐴))
9		omelon 9542	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
10	9	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ω ∈ On)
11		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐵 ∈ On)
12	10, 11	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ∈ On ∧ 𝐵 ∈ On))
13		peano1 7822	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
14		oen0 8504	. . . . . . . . . . . . 13 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
15	12, 13, 14	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ (ω ↑o 𝐵))
16		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 = (ω ↑o 𝐵))
17	15, 16	eleqtrrd 2831	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ 𝐶)
18		fconst6g 6713	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝐶 → (𝐴 × {∅}):𝐴⟶𝐶)
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (𝐴 × {∅}):𝐴⟶𝐶)
20	19	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}):𝐴⟶𝐶)
21		oecl 8455	. . . . . . . . . . . . 13 ⊢ ((ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑o 𝐵) ∈ On)
22	9, 11, 21	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ↑o 𝐵) ∈ On)
23	16, 22	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 ∈ On)
24	23	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐶 ∈ On)
25		simpl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐴 ∈ 𝑉)
26	24, 25	elmapd 8767	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) ↔ (𝐴 × {∅}):𝐴⟶𝐶))
27	20, 26	mpbird 257	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))
28	27	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))
29		ovres 7515	. . . . . . . . . 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 481	. . . . . . . . 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 8776	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝐶 ↑m 𝐴) → ℎ:𝐴⟶𝐶)
32	31	adantr 480	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → ℎ:𝐴⟶𝐶)
33	32	ffnd 6653	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → ℎ Fn 𝐴)
34	33	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → ℎ Fn 𝐴)
35		elmapi 8776	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) → (𝐴 × {∅}):𝐴⟶𝐶)
36	35	adantl 481	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (𝐴 × {∅}):𝐴⟶𝐶)
37	36	ffnd 6653	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
38	37	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → (𝐴 × {∅}) Fn 𝐴)
39	25	adantr 480	. . . . . . . . . . 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 7626	. . . . . . . . . 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 8792	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝐶 ↑m 𝐴) → ℎ Fn 𝐴)
42		elmapfn 8792	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) → (𝐴 × {∅}) Fn 𝐴)
43	41, 42	anim12i 613	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (ℎ Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
44	43	adantl 481	. . . . . . . . . . . 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 615	. . . . . . . . . . . 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 7630	. . . . . . . . . . . 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 685	. . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
49		fvconst2g 7138	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ ω ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
50	13, 48, 49	sylancr 587	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
51	50	oveq2d 7365	. . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → 𝐶 ∈ On)
53	52	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ On)
54		onss 7721	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
55	53, 54	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ On)
56	31	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → ℎ:𝐴⟶𝐶)
57	56	ffvelcdmda 7018	. . . . . . . . . . . . 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 3936	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (ℎ‘𝑎) ∈ On)
59		oa0 8434	. . . . . . . . . . . 12 ⊢ ((ℎ‘𝑎) ∈ On → ((ℎ‘𝑎) +o ∅) = (ℎ‘𝑎))
60	58, 59	syl 17	. . . . . . . . . . 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 2768	. . . . . . . . . 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 6968	. . . . . . . . 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 2765	. . . . . . . 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 456	. . . . . . 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 516	. . . . . 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 7357	. . . . . . 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 3600	. . . . . 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 17	. . . . 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 511	. . . 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 7356	. . . . . . 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 2740	. . . . . 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 3153	. . . . 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 3577	. . . 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 17	. . 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 3121	. 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 7523	. 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 583	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))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  {csn 4577   × cxp 5617   ↾ cres 5621  Oncon0 6307   Fn wfn 6477  ⟶wf 6478  –onto→wfo 6480  ‘cfv 6482  (class class class)co 7349   ∘_f cof 7611  ωcom 7799   +_o coa 8385   ↑_o coe 8387   ↑_m cmap 8753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-omul 8393  df-oexp 8394  df-map 8755

This theorem is referenced by: (None)