Theorem naddcnffo 42690

Description: Addition of Cantor normal forms is a function onto Cantor normal forms. (Contributed by RP, 2-Jan-2025.)

Assertion

Ref	Expression
naddcnffo	⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆)

Proof of Theorem naddcnffo

Dummy variables 𝑓 𝑔 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		naddcnff 42688	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
2		simpr 484	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ 𝑆)
3		peano1 7876	. . . . . . . . 9 ⊢ ∅ ∈ ω
4		fconst6g 6774	. . . . . . . . 9 ⊢ (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
5	3, 4	mp1i 13	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
6		simpl 482	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
7	3	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∅ ∈ ω)
8	6, 7	fczfsuppd 9383	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
9		simpr 484	. . . . . . . . . 10 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
10	9	eleq2d 2813	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
11		eqid 2726	. . . . . . . . . 10 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
12		omelon 9643	. . . . . . . . . . 11 ⊢ ω ∈ On
13	12	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
14	11, 13, 6	cantnfs 9663	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
15	10, 14	bitrd 279	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
16	5, 8, 15	mpbir2and 710	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
17	16	adantr 480	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → (𝑋 × {∅}) ∈ 𝑆)
18		simpl 482	. . . . . . . . . 10 ⊢ ((𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑓 ∈ 𝑆)
19	18	adantl 481	. . . . . . . . 9 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 ∈ 𝑆)
20		simpr 484	. . . . . . . . . 10 ⊢ ((𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝑋 × {∅}) ∈ 𝑆)
21	20	adantl 481	. . . . . . . . 9 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑋 × {∅}) ∈ 𝑆)
22	19, 21	ovresd 7571	. . . . . . . 8 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})) = (𝑓 ∘f +o (𝑋 × {∅})))
23	9	eleq2d 2813	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 ↔ 𝑓 ∈ dom (ω CNF 𝑋)))
24	11, 13, 6	cantnfs 9663	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ dom (ω CNF 𝑋) ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
25	23, 24	bitrd 279	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
26	25	biimpd 228	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 → (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
27		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅) → 𝑓:𝑋⟶ω)
28	18, 26, 27	syl56 36	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑓:𝑋⟶ω))
29	28	imp 406	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓:𝑋⟶ω)
30	29	ffnd 6712	. . . . . . . . . 10 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 Fn 𝑋)
31		fnconstg 6773	. . . . . . . . . . 11 ⊢ (∅ ∈ ω → (𝑋 × {∅}) Fn 𝑋)
32	3, 31	mp1i 13	. . . . . . . . . 10 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑋 × {∅}) Fn 𝑋)
33	6	adantr 480	. . . . . . . . . 10 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑋 ∈ On)
34		inidm 4213	. . . . . . . . . 10 ⊢ (𝑋 ∩ 𝑋) = 𝑋
35	30, 32, 33, 33, 34	offn 7680	. . . . . . . . 9 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓 ∘f +o (𝑋 × {∅})) Fn 𝑋)
36	30	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝑓 Fn 𝑋)
37	3, 31	mp1i 13	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {∅}) Fn 𝑋)
38		simplll 772	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ On)
39		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
40		fnfvof 7684	. . . . . . . . . . 11 ⊢ (((𝑓 Fn 𝑋 ∧ (𝑋 × {∅}) Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → ((𝑓 ∘f +o (𝑋 × {∅}))‘𝑥) = ((𝑓‘𝑥) +o ((𝑋 × {∅})‘𝑥)))
41	36, 37, 38, 39, 40	syl22anc 836	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝑓 ∘f +o (𝑋 × {∅}))‘𝑥) = ((𝑓‘𝑥) +o ((𝑋 × {∅})‘𝑥)))
42	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ∅ ∈ ω)
43		fvconst2g 7199	. . . . . . . . . . . 12 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
44	42, 39, 43	syl2anc 583	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
45	44	oveq2d 7421	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) +o ((𝑋 × {∅})‘𝑥)) = ((𝑓‘𝑥) +o ∅))
46	29	ffvelcdmda 7080	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ ω)
47		nnon 7858	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ ω → (𝑓‘𝑥) ∈ On)
48		oa0 8517	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ On → ((𝑓‘𝑥) +o ∅) = (𝑓‘𝑥))
49	46, 47, 48	3syl 18	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) +o ∅) = (𝑓‘𝑥))
50	41, 45, 49	3eqtrd 2770	. . . . . . . . 9 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝑓 ∘f +o (𝑋 × {∅}))‘𝑥) = (𝑓‘𝑥))
51	35, 30, 50	eqfnfvd 7029	. . . . . . . 8 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓 ∘f +o (𝑋 × {∅})) = 𝑓)
52	22, 51	eqtr2d 2767	. . . . . . 7 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
53	52	expr 456	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → ((𝑋 × {∅}) ∈ 𝑆 → 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))))
54	17, 53	jcai 516	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → ((𝑋 × {∅}) ∈ 𝑆 ∧ 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))))
55		oveq2 7413	. . . . . 6 ⊢ (𝑧 = (𝑋 × {∅}) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
56	55	rspceeqv 3628	. . . . 5 ⊢ (((𝑋 × {∅}) ∈ 𝑆 ∧ 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))) → ∃𝑧 ∈ 𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
57	54, 56	syl 17	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → ∃𝑧 ∈ 𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
58		oveq1 7412	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
59	58	eqeq2d 2737	. . . . . 6 ⊢ (𝑔 = 𝑓 → (𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
60	59	rexbidv 3172	. . . . 5 ⊢ (𝑔 = 𝑓 → (∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ ∃𝑧 ∈ 𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
61	60	rspcev 3606	. . . 4 ⊢ ((𝑓 ∈ 𝑆 ∧ ∃𝑧 ∈ 𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)) → ∃𝑔 ∈ 𝑆 ∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
62	2, 57, 61	syl2anc 583	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → ∃𝑔 ∈ 𝑆 ∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
63	62	ralrimiva 3140	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓 ∈ 𝑆 ∃𝑔 ∈ 𝑆 ∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
64		foov 7578	. 2 ⊢ (( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆 ↔ (( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆 ∧ ∀𝑓 ∈ 𝑆 ∃𝑔 ∈ 𝑆 ∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
65	1, 63, 64	sylanbrc 582	1 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ∃wrex 3064  ∅c0 4317  {csn 4623   class class class wbr 5141   × cxp 5667  dom cdm 5669   ↾ cres 5671  Oncon0 6358   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7405   ∘_f cof 7665  ωcom 7852   +_o coa 8464   finSupp cfsupp 9363   CNF ccnf 9658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-seqom 8449  df-1o 8467  df-oadd 8471  df-map 8824  df-en 8942  df-fin 8945  df-fsupp 9364  df-cnf 9659

This theorem is referenced by: (None)