Theorem cantnfrescl 9723

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 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 ∈ 𝐴)
6	5	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ (𝐷 ∖ 𝐵)𝑋 ∈ 𝐴)
7	1, 6	raldifeq 4503	. . . 4 ⊢ (𝜑 → (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ ∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴))
8		eqid 2737	. . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋)
9	8	fmpt 7137	. . . 4 ⊢ (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴)
10		eqid 2737	. . . . 5 ⊢ (𝑛 ∈ 𝐷 ↦ 𝑋) = (𝑛 ∈ 𝐷 ↦ 𝑋)
11	10	fmpt 7137	. . . 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 7251	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V)
15		funmpt 6612	. . . . . 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 7251	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V)
19		funmpt 6612	. . . . . 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 8221	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅))
23		suppeqfsuppbi 9426	. . . 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 9713	. 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 9713	. 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 1539   ∈ wcel 2108  ∀wral 3061  Vcvv 3481   ∖ cdif 3963   ⊆ wss 3966  ∅c0 4342   class class class wbr 5151   ↦ cmpt 5234  dom cdm 5693  Oncon0 6392  Fun wfun 6563  ⟶wf 6565  (class class class)co 7438   supp csupp 8193   finSupp cfsupp 9408   CNF ccnf 9708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-pred 6329  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-supp 8194  df-frecs 8314  df-wrecs 8345  df-recs 8419  df-rdg 8458  df-seqom 8496  df-map 8876  df-fsupp 9409  df-cnf 9709

This theorem is referenced by:  cantnfres  9724