Theorem cantnfrescl 9572

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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → ∅ ∈ 𝐴)
5	2, 4	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 ∈ 𝐴)
6	5	ralrimiva 3121	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ (𝐷 ∖ 𝐵)𝑋 ∈ 𝐴)
7	1, 6	raldifeq 4445	. . . 4 ⊢ (𝜑 → (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ ∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴))
8		eqid 2729	. . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
9	8	fmpt 7044	. . . 4 ⊢ (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴)
10		eqid 2729	. . . . 5 ⊢ (𝑛 ∈ 𝐷 ↦ 𝑋) = (𝑛 ∈ 𝐷 ↦ 𝑋)
11	10	fmpt 7044	. . . 4 ⊢ (∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴)
12	7, 9, 11	3bitr3g 313	. . 3 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴))
13		cantnfs.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ On)
14	13	mptexd 7160	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V)
15		funmpt 6520	. . . . . 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 7160	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V)
19		funmpt 6520	. . . . . 6 ⊢ Fun (𝑛 ∈ 𝐷 ↦ 𝑋)
20	18, 19	jctir 520	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋)))
21	14, 16, 20	jca31 514	. . . 4 ⊢ (𝜑 → (((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐵 ↦ 𝑋)) ∧ ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋))))
22	17, 1, 2	extmptsuppeq 8121	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))
23		suppeqfsuppbi 9269	. . . 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 65	. . 3 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) finSupp ∅ ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅))
25	12, 24	anbi12d 632	. 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 9562	. 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 9562	. 2 ⊢ (𝜑 → ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇 ↔ ((𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴 ∧ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅)))
31	25, 28, 30	3bitr4d 311	1 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4284   class class class wbr 5092   ↦ cmpt 5173  dom cdm 5619  Oncon0 6307  Fun wfun 6476  ⟶wf 6478  (class class class)co 7349   supp csupp 8093   finSupp cfsupp 9251   CNF ccnf 9557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-seqom 8370  df-map 8755  df-fsupp 9252  df-cnf 9558

This theorem is referenced by:  cantnfres  9573