Theorem ofoafo 43938

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 4180	. . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	eqcomi 2773	. . . . 5 ⊢ 𝐴 = (𝐴 ∩ 𝐴)
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = (𝐴 ∩ 𝐴))
5	1, 1, 4	3jca 1142	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 = (𝐴 ∩ 𝐴)))
6		ofoaf 43937	. . 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 589	. 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 488	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → ℎ ∈ (𝐶 ↑m 𝐴))
9		omelon 9603	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
10	9	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ω ∈ On)
11		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐵 ∈ On)
12	10, 11	jca 519	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ∈ On ∧ 𝐵 ∈ On))
13		peano1 7871	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
14		oen0 8558	. . . . . . . . . . . . 13 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐵))
15	12, 13, 14	sylancl 595	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ (ω ↑o 𝐵))
16		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 = (ω ↑o 𝐵))
17	15, 16	eleqtrrd 2867	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → ∅ ∈ 𝐶)
18		fconst6g 6755	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝐶 → (𝐴 × {∅}):𝐴⟶𝐶)
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (𝐴 × {∅}):𝐴⟶𝐶)
20	19	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}):𝐴⟶𝐶)
21		oecl 8508	. . . . . . . . . . . . 13 ⊢ ((ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑o 𝐵) ∈ On)
22	9, 11, 21	sylancr 596	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → (ω ↑o 𝐵) ∈ On)
23	16, 22	eqeltrd 2864	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵)) → 𝐶 ∈ On)
24	23	adantl 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐶 ∈ On)
25		simpl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → 𝐴 ∈ 𝑉)
26	24, 25	elmapd 8823	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) ↔ (𝐴 × {∅}):𝐴⟶𝐶))
27	20, 26	mpbird 259	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))
28	27	adantr 484	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ ℎ ∈ (𝐶 ↑m 𝐴)) → (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))
29		ovres 7564	. . . . . . . . . 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 485	. . . . . . . . 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 8832	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝐶 ↑m 𝐴) → ℎ:𝐴⟶𝐶)
32	31	adantr 484	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → ℎ:𝐴⟶𝐶)
33	32	ffnd 6694	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → ℎ Fn 𝐴)
34	33	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → ℎ Fn 𝐴)
35		elmapi 8832	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) → (𝐴 × {∅}):𝐴⟶𝐶)
36	35	adantl 485	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (𝐴 × {∅}):𝐴⟶𝐶)
37	36	ffnd 6694	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
38	37	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → (𝐴 × {∅}) Fn 𝐴)
39	25	adantr 484	. . . . . . . . . . 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 7675	. . . . . . . . . 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 8848	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ (𝐶 ↑m 𝐴) → ℎ Fn 𝐴)
42		elmapfn 8848	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴) → (𝐴 × {∅}) Fn 𝐴)
43	41, 42	anim12i 622	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴)) → (ℎ Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
44	43	adantl 485	. . . . . . . . . . . 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 624	. . . . . . . . . . . 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 7679	. . . . . . . . . . . 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 695	. . . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
49		fvconst2g 7188	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ ω ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
50	13, 48, 49	sylancr 596	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
51	50	oveq2d 7414	. . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → 𝐶 ∈ On)
53	52	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ On)
54		onss 7770	. . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) → ℎ:𝐴⟶𝐶)
57	56	ffvelcdmda 7067	. . . . . . . . . . . . 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 3939	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (ℎ ∈ (𝐶 ↑m 𝐴) ∧ (𝐴 × {∅}) ∈ (𝐶 ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (ℎ‘𝑎) ∈ On)
59		oa0 8487	. . . . . . . . . . . 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 2803	. . . . . . . . . 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 7016	. . . . . . . . 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 2800	. . . . . . . 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 460	. . . . . . 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 524	. . . . . 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 7406	. . . . . . 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 3606	. . . . . 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 519	. . . 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 7405	. . . . . . 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 2775	. . . . . 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 3583	. . . 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 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 7572	. 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 592	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 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  {csn 4584   × cxp 5647   ↾ cres 5651  Oncon0 6348   Fn wfn 6518  ⟶wf 6519  –onto→wfo 6521  ‘cfv 6523  (class class class)co 7398   ∘_f cof 7660  ωcom 7848   +_o coa 8436   ↑_o coe 8438   ↑_m cmap 8810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-of 7662  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-omul 8444  df-oexp 8445  df-map 8812

This theorem is referenced by: (None)