Theorem mapfi 7216

Description: Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)

Assertion

Ref	Expression
mapfi	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ↑𝑚 𝐵) ∈ Fin)

Proof of Theorem mapfi

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6060	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ↑𝑚 𝑤) = (𝐴 ↑𝑚 ∅))
2	1	eleq1d 2303	. 2 ⊢ (𝑤 = ∅ → ((𝐴 ↑𝑚 𝑤) ∈ Fin ↔ (𝐴 ↑𝑚 ∅) ∈ Fin))
3		oveq2 6060	. . 3 ⊢ (𝑤 = 𝑦 → (𝐴 ↑𝑚 𝑤) = (𝐴 ↑𝑚 𝑦))
4	3	eleq1d 2303	. 2 ⊢ (𝑤 = 𝑦 → ((𝐴 ↑𝑚 𝑤) ∈ Fin ↔ (𝐴 ↑𝑚 𝑦) ∈ Fin))
5		oveq2 6060	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐴 ↑𝑚 𝑤) = (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})))
6	5	eleq1d 2303	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐴 ↑𝑚 𝑤) ∈ Fin ↔ (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ∈ Fin))
7		oveq2 6060	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ↑𝑚 𝑤) = (𝐴 ↑𝑚 𝐵))
8	7	eleq1d 2303	. 2 ⊢ (𝑤 = 𝐵 → ((𝐴 ↑𝑚 𝑤) ∈ Fin ↔ (𝐴 ↑𝑚 𝐵) ∈ Fin))
9		mapdm0 6899	. . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 ↑𝑚 ∅) = {∅})
10		0ex 4239	. . . . 5 ⊢ ∅ ∈ V
11		snfig 7058	. . . . 5 ⊢ (∅ ∈ V → {∅} ∈ Fin)
12	10, 11	ax-mp 5	. . . 4 ⊢ {∅} ∈ Fin
13	9, 12	eqeltrdi 2325	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ↑𝑚 ∅) ∈ Fin)
14	13	adantr 276	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ↑𝑚 ∅) ∈ Fin)
15		simpr 110	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → (𝐴 ↑𝑚 𝑦) ∈ Fin)
16		simp-4l 543	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → 𝐴 ∈ Fin)
17		vex 2818	. . . . . . . 8 ⊢ 𝑧 ∈ V
18	17	a1i 9	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → 𝑧 ∈ V)
19	16, 18	mapsnend 7054	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴)
20		enfii 7131	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴) → (𝐴 ↑𝑚 {𝑧}) ∈ Fin)
21	16, 19, 20	syl2anc 411	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → (𝐴 ↑𝑚 {𝑧}) ∈ Fin)
22		xpfi 7194	. . . . 5 ⊢ (((𝐴 ↑𝑚 𝑦) ∈ Fin ∧ (𝐴 ↑𝑚 {𝑧}) ∈ Fin) → ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ∈ Fin)
23	15, 21, 22	syl2anc 411	. . . 4 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ∈ Fin)
24		vex 2818	. . . . . 6 ⊢ 𝑦 ∈ V
25	24	a1i 9	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → 𝑦 ∈ V)
26	17	snex 4300	. . . . . 6 ⊢ {𝑧} ∈ V
27	26	a1i 9	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → {𝑧} ∈ V)
28		simplrr 538	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → 𝑧 ∈ (𝐵 ∖ 𝑦))
29	28	eldifbd 3225	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
30		disjsn 3753	. . . . . 6 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
31	29, 30	sylibr 134	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → (𝑦 ∩ {𝑧}) = ∅)
32		mapunen 7106	. . . . 5 ⊢ (((𝑦 ∈ V ∧ {𝑧} ∈ V ∧ 𝐴 ∈ Fin) ∧ (𝑦 ∩ {𝑧}) = ∅) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})))
33	25, 27, 16, 31, 32	syl31anc 1277	. . . 4 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})))
34		enfii 7131	. . . 4 ⊢ ((((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ∈ Fin ∧ (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧}))) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ∈ Fin)
35	23, 33, 34	syl2anc 411	. . 3 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ∈ Fin)
36	35	ex 115	. 2 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) → ((𝐴 ↑𝑚 𝑦) ∈ Fin → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ∈ Fin))
37		simpr 110	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → 𝐵 ∈ Fin)
38	2, 4, 6, 8, 14, 36, 37	findcard2sd 7151	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ↑𝑚 𝐵) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ∖ cdif 3210   ∪ cun 3211   ∩ cin 3212   ⊆ wss 3213  ∅c0 3510  {csn 3691   class class class wbr 4111   × cxp 4749  (class class class)co 6052   ↑_𝑚 cmap 6884   ≈ cen 6975  Fincfn 6977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-er 6769  df-map 6886  df-en 6978  df-fin 6980

This theorem is referenced by:  fipwfi  7274  hashmap  11196  hashpwfi  11197