Theorem mapfi 7254

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 6086	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ↑𝑚 𝑤) = (𝐴 ↑𝑚 ∅))
2	1	eleq1d 2307	. 2 ⊢ (𝑤 = ∅ → ((𝐴 ↑𝑚 𝑤) ∈ Fin ↔ (𝐴 ↑𝑚 ∅) ∈ Fin))
3		oveq2 6086	. . 3 ⊢ (𝑤 = 𝑦 → (𝐴 ↑𝑚 𝑤) = (𝐴 ↑𝑚 𝑦))
4	3	eleq1d 2307	. 2 ⊢ (𝑤 = 𝑦 → ((𝐴 ↑𝑚 𝑤) ∈ Fin ↔ (𝐴 ↑𝑚 𝑦) ∈ Fin))
5		oveq2 6086	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐴 ↑𝑚 𝑤) = (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})))
6	5	eleq1d 2307	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐴 ↑𝑚 𝑤) ∈ Fin ↔ (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ∈ Fin))
7		oveq2 6086	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ↑𝑚 𝑤) = (𝐴 ↑𝑚 𝐵))
8	7	eleq1d 2307	. 2 ⊢ (𝑤 = 𝐵 → ((𝐴 ↑𝑚 𝑤) ∈ Fin ↔ (𝐴 ↑𝑚 𝐵) ∈ Fin))
9		mapdm0 6930	. . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 ↑𝑚 ∅) = {∅})
10		0ex 4258	. . . . 5 ⊢ ∅ ∈ V
11		snfig 7096	. . . . 5 ⊢ (∅ ∈ V → {∅} ∈ Fin)
12	10, 11	ax-mp 5	. . . 4 ⊢ {∅} ∈ Fin
13	9, 12	eqeltrdi 2329	. . 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 547	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → 𝐴 ∈ Fin)
17		vex 2824	. . . . . . . 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 7092	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴)
20		enfii 7169	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴) → (𝐴 ↑𝑚 {𝑧}) ∈ Fin)
21	16, 19, 20	syl2anc 415	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → (𝐴 ↑𝑚 {𝑧}) ∈ Fin)
22		xpfi 7232	. . . . 5 ⊢ (((𝐴 ↑𝑚 𝑦) ∈ Fin ∧ (𝐴 ↑𝑚 {𝑧}) ∈ Fin) → ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ∈ Fin)
23	15, 21, 22	syl2anc 415	. . . 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 2824	. . . . . 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 4320	. . . . . 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 542	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → 𝑧 ∈ (𝐵 ∖ 𝑦))
29	28	eldifbd 3232	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
30		disjsn 3770	. . . . . 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 7144	. . . . 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 1281	. . . 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 7169	. . . 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 415	. . 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 7189	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ↑𝑚 𝐵) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∖ cdif 3217   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  {csn 3708   class class class wbr 4128   × cxp 4770  (class class class)co 6078   ↑_𝑚 cmap 6915   ≈ cen 7013  Fincfn 7015

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-1o 6680  df-er 6800  df-map 6917  df-en 7016  df-fin 7018

This theorem is referenced by:  f1setfi  7310  fipwfi  7314  hashmap  11249  hashpwfi  11250