Theorem naddcnffo 43606

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 43604	. 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 7831	. . . . . . . . 9 ⊢ ∅ ∈ ω
4		fconst6g 6723	. . . . . . . . 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 9289	. . . . . . . 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 2822	. . . . . . . . 9 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
11		eqid 2736	. . . . . . . . . 10 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
12		omelon 9555	. . . . . . . . . . 11 ⊢ ω ∈ On
13	12	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
14	11, 13, 6	cantnfs 9575	. . . . . . . . 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 713	. . . . . . 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 7525	. . . . . . . 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 2822	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑓 ∈ 𝑆 ↔ 𝑓 ∈ dom (ω CNF 𝑋)))
24	11, 13, 6	cantnfs 9575	. . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . 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 6663	. . . . . . . . . 10 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → 𝑓 Fn 𝑋)
31		fnconstg 6722	. . . . . . . . . . 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 4179	. . . . . . . . . 10 ⊢ (𝑋 ∩ 𝑋) = 𝑋
35	30, 32, 33, 33, 34	offn 7635	. . . . . . . . 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 774	. . . . . . . . . . 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 7639	. . . . . . . . . . 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 838	. . . . . . . . . 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 7148	. . . . . . . . . . . 12 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
44	42, 39, 43	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
45	44	oveq2d 7374	. . . . . . . . . 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 7029	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ ω)
47		nnon 7814	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ ω → (𝑓‘𝑥) ∈ On)
48		oa0 8443	. . . . . . . . . . 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 2775	. . . . . . . . 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 6979	. . . . . . . 8 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝑓 ∈ 𝑆 ∧ (𝑋 × {∅}) ∈ 𝑆)) → (𝑓 ∘f +o (𝑋 × {∅})) = 𝑓)
52	22, 51	eqtr2d 2772	. . . . . . 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 7366	. . . . . 6 ⊢ (𝑧 = (𝑋 × {∅}) → (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))(𝑋 × {∅})))
56	55	rspceeqv 3599	. . . . 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 7365	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
59	58	eqeq2d 2747	. . . . . 6 ⊢ (𝑔 = 𝑓 → (𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
60	59	rexbidv 3160	. . . . 5 ⊢ (𝑔 = 𝑓 → (∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧) ↔ ∃𝑧 ∈ 𝑆 𝑓 = (𝑓( ∘f +o ↾ (𝑆 × 𝑆))𝑧)))
61	60	rspcev 3576	. . . 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 584	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝑓 ∈ 𝑆) → ∃𝑔 ∈ 𝑆 ∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
63	62	ralrimiva 3128	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∀𝑓 ∈ 𝑆 ∃𝑔 ∈ 𝑆 ∃𝑧 ∈ 𝑆 𝑓 = (𝑔( ∘f +o ↾ (𝑆 × 𝑆))𝑧))
64		foov 7532	. 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 583	1 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  ∅c0 4285  {csn 4580   class class class wbr 5098   × cxp 5622  dom cdm 5624   ↾ cres 5626  Oncon0 6317   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358   ∘_f cof 7620  ωcom 7808   +_o coa 8394   finSupp cfsupp 9264   CNF ccnf 9570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-seqom 8379  df-1o 8397  df-oadd 8401  df-map 8765  df-en 8884  df-fin 8887  df-fsupp 9265  df-cnf 9571

This theorem is referenced by: (None)