Theorem naddcnffo 42858

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 42856	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
2		simpr 483	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ 𝑆)
3		peano1 7892	. . . . . . . . 9 ⊢ ∅ ∈ ω
4		fconst6g 6781	. . . . . . . . 9 ⊢ (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
5	3, 4	mp1i 13	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
6		simpl 481	. . . . . . . . 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 9409	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
9		simpr 483	. . . . . . . . . 10 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
10	9	eleq2d 2811	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
11		eqid 2725	. . . . . . . . . 10 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
12		omelon 9669	. . . . . . . . . . 11 ⊢ ω ∈ On
13	12	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
14	11, 13, 6	cantnfs 9689	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
15	10, 14	bitrd 278	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
16	5, 8, 15	mpbir2and 711	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
17	16	adantr 479	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → (𝑋 × {∅}) ∈ 𝑆)
18		simpl 481	. . . . . . . . . 10 ⊢ ((𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑓 ∈ 𝑆)
19	18	adantl 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 ∈ 𝑆)
20		simpr 483	. . . . . . . . . 10 ⊢ ((𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝑋 × {∅}) ∈ 𝑆)
21	20	adantl 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑋 × {∅}) ∈ 𝑆)
22	19, 21	ovresd 7585	. . . . . . . 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 2811	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 ↔ 𝑓 ∈ dom (ω CNF 𝑋)))
24	11, 13, 6	cantnfs 9689	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ dom (ω CNF 𝑋) ↔ (𝑓:𝑋⟶ω ∧ 𝑓 finSupp ∅)))
25	23, 24	bitrd 278	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 405	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓:𝑋⟶ω)
30	29	ffnd 6718	. . . . . . . . . 10 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 Fn 𝑋)
31		fnconstg 6780	. . . . . . . . . . 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 479	. . . . . . . . . 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 7695	. . . . . . . . 9 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓 ∘f +o (𝑋 × {∅})) Fn 𝑋)
36	30	adantr 479	. . . . . . . . . . 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 773	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ On)
39		simpr 483	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
40		fnfvof 7699	. . . . . . . . . . 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 837	. . . . . . . . . 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 7210	. . . . . . . . . . . 12 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
44	42, 39, 43	syl2anc 582	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
45	44	oveq2d 7432	. . . . . . . . . 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 7089	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ ω)
47		nnon 7874	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ ω → (𝑓‘𝑥) ∈ On)
48		oa0 8535	. . . . . . . . . . 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 2769	. . . . . . . . 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 7038	. . . . . . . 8 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓 ∘f +o (𝑋 × {∅})) = 𝑓)
52	22, 51	eqtr2d 2766	. . . . . . 7 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
53	52	expr 455	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → ((𝑋 × {∅}) ∈ 𝑆 → 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))))
54	17, 53	jcai 515	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → ((𝑋 × {∅}) ∈ 𝑆 ∧ 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅}))))
55		oveq2 7424	. . . . . 6 ⊢ (𝑧 = (𝑋 × {∅}) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
56	55	rspceeqv 3623	. . . . 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 7423	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
59	58	eqeq2d 2736	. . . . . 6 ⊢ (𝑔 = 𝑓 → (𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
60	59	rexbidv 3169	. . . . 5 ⊢ (𝑔 = 𝑓 → (∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ ∃𝑧 ∈ 𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
61	60	rspcev 3601	. . . 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 582	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → ∃𝑔 ∈ 𝑆 ∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
63	62	ralrimiva 3136	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓 ∈ 𝑆 ∃𝑔 ∈ 𝑆 ∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
64		foov 7592	. 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 581	1 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051  ∃wrex 3060  ∅c0 4318  {csn 4624   class class class wbr 5143   × cxp 5670  dom cdm 5672   ↾ cres 5674  Oncon0 6364   Fn wfn 6538  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7416   ∘_f cof 7680  ωcom 7868   +_o coa 8482   finSupp cfsupp 9385   CNF ccnf 9684

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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-supp 8164  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-seqom 8467  df-1o 8485  df-oadd 8489  df-map 8845  df-en 8963  df-fin 8966  df-fsupp 9386  df-cnf 9685

This theorem is referenced by: (None)