Theorem ofoafo 43784

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 4167	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	eqcomi 2745	. . . . 5 ⊢ 𝐴 = (𝐴 ∩ 𝐴)
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = (𝐴 ∩ 𝐴))
5	1, 1, 4	3jca 1129	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 = (𝐴 ∩ 𝐴)))
6		ofoaf 43783	. . 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 581	. 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 9567	. . . . . . . . . . . . . . 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 7840	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
14		oen0 8522	. . . . . . . . . . . . 13 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
15	12, 13, 14	sylancl 587	. . . . . . . . . . . 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 2839	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ 𝐶)
18		fconst6g 6729	. . . . . . . . . . 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 8472	. . . . . . . . . . . . 13 ⊢ ((ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑o 𝐵) ∈ On)
22	9, 11, 21	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ↑o 𝐵) ∈ On)
23	16, 22	eqeltrd 2836	. . . . . . . . . . 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 8787	. . . . . . . . 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 7533	. . . . . . . . . 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 8796	. . . . . . . . . . . . . 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 6669	. . . . . . . . . . . 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 8796	. . . . . . . . . . . . . 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 6669	. . . . . . . . . . . 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 7644	. . . . . . . . . 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 8812	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝐶 ↑m 𝐴) → ℎ Fn 𝐴)
42		elmapfn 8812	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) → (𝐴 × {∅}) Fn 𝐴)
43	41, 42	anim12i 614	. . . . . . . . . . . . 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 616	. . . . . . . . . . . 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 7648	. . . . . . . . . . . 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 686	. . . . . . . . . . 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 7157	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ ω ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
50	13, 48, 49	sylancr 588	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
51	50	oveq2d 7383	. . . . . . . . . . 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 7739	. . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → ℎ:𝐴⟶𝐶)
57	56	ffvelcdmda 7036	. . . . . . . . . . . . 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 3922	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (ℎ‘𝑎) ∈ On)
59		oa0 8451	. . . . . . . . . . . 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 2775	. . . . . . . . . 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 6986	. . . . . . . . 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 2772	. . . . . . . 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 7375	. . . . . . 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 3587	. . . . . 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 7374	. . . . . . 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 2747	. . . . . 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 3161	. . . . 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 3564	. . . 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 3129	. 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 7541	. 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 584	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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  {csn 4567   × cxp 5629   ↾ cres 5633  Oncon0 6323   Fn wfn 6493  ⟶wf 6494  –onto→wfo 6496  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629  ωcom 7817   +_o coa 8402   ↑_o coe 8404   ↑_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  ax-inf2 9562

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-1o 8405  df-2o 8406  df-oadd 8409  df-omul 8410  df-oexp 8411  df-map 8775

This theorem is referenced by: (None)