Theorem cantnfrescl 9641

Description: A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfrescl.d	⊢ (𝜑 → 𝐷 ∈ On)
cantnfrescl.b	⊢ (𝜑 → 𝐵 ⊆ 𝐷)
cantnfrescl.x	⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)
cantnfrescl.a	⊢ (𝜑 → ∅ ∈ 𝐴)
cantnfrescl.t	⊢ 𝑇 = dom (𝐴 CNF 𝐷)

Assertion

Ref	Expression
cantnfrescl	⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))

Distinct variable groups:   𝐵,𝑛   𝐷,𝑛   𝐴,𝑛   𝜑,𝑛

Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑋(𝑛)

Proof of Theorem cantnfrescl

Step	Hyp	Ref	Expression
1		cantnfrescl.b	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐷)
2		cantnfrescl.x	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)
3		cantnfrescl.a	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐴)
4	3	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → ∅ ∈ 𝐴)
5	2, 4	eqeltrd 2863	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 ∈ 𝐴)
6	5	ralrimiva 3157	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ (𝐷 ∖ 𝐵)𝑋 ∈ 𝐴)
7	1, 6	raldifeq 4454	. . . 4 ⊢ (𝜑 → (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ ∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴))
8		eqid 2763	. . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
9	8	fmpt 7105	. . . 4 ⊢ (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴)
10		eqid 2763	. . . . 5 ⊢ (𝑛 ∈ 𝐷 ↦ 𝑋) = (𝑛 ∈ 𝐷 ↦ 𝑋)
11	10	fmpt 7105	. . . 4 ⊢ (∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴)
12	7, 9, 11	3bitr3g 316	. . 3 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴))
13		cantnfs.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ On)
14	13	mptexd 7222	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V)
15		funmpt 6574	. . . . . 6 ⊢ Fun (𝑛 ∈ 𝐵 ↦ 𝑋)
16	15	a1i 11	. . . . 5 ⊢ (𝜑 → Fun (𝑛 ∈ 𝐵 ↦ 𝑋))
17		cantnfrescl.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ On)
18	17	mptexd 7222	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V)
19		funmpt 6574	. . . . . 6 ⊢ Fun (𝑛 ∈ 𝐷 ↦ 𝑋)
20	18, 19	jctir 529	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋)))
21	14, 16, 20	jca31 523	. . . 4 ⊢ (𝜑 → (((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐵 ↦ 𝑋)) ∧ ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋))))
22	17, 1, 2	extmptsuppeq 8180	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))
23		suppeqfsuppbi 9335	. . . 4 ⊢ ((((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐵 ↦ 𝑋)) ∧ ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋))) → (((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅) → ((𝑛 ∈ 𝐵 ↦ 𝑋) finSupp ∅ ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅)))
24	21, 22, 23	sylc 66	. . 3 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) finSupp ∅ ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅))
25	12, 24	anbi12d 643	. 2 ⊢ (𝜑 → (((𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴 ∧ (𝑛 ∈ 𝐵 ↦ 𝑋) finSupp ∅) ↔ ((𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴 ∧ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅)))
26		cantnfs.s	. . 3 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
27		cantnfs.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
28	26, 27, 13	cantnfs 9631	. 2 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ ((𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴 ∧ (𝑛 ∈ 𝐵 ↦ 𝑋) finSupp ∅)))
29		cantnfrescl.t	. . 3 ⊢ 𝑇 = dom (𝐴 CNF 𝐷)
30	29, 27, 17	cantnfs 9631	. 2 ⊢ (𝜑 → ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇 ↔ ((𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴 ∧ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅)))
31	25, 28, 30	3bitr4d 314	1 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5661  Oncon0 6360  Fun wfun 6530  ⟶wf 6532  (class class class)co 7410   supp csupp 8152   finSupp cfsupp 9317   CNF ccnf 9626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-seqom 8431  df-map 8822  df-fsupp 9318  df-cnf 9627

This theorem is referenced by:  cantnfres  9642