Theorem mpct 40887

Description: The exponentiation of a countable set to a finite set is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
mpct.a	⊢ (𝜑 → 𝐴 ≼ ω)
mpct.b	⊢ (𝜑 → 𝐵 ∈ Fin)

Assertion

Ref	Expression
mpct	⊢ (𝜑 → (𝐴 ↑𝑚 𝐵) ≼ ω)

Proof of Theorem mpct

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

Step	Hyp	Ref	Expression
1		oveq2 6984	. . 3 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 ∅))
2	1	breq1d 4939	. 2 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑚 𝑥) ≼ ω ↔ (𝐴 ↑𝑚 ∅) ≼ ω))
3		oveq2 6984	. . 3 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 𝑦))
4	3	breq1d 4939	. 2 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑚 𝑥) ≼ ω ↔ (𝐴 ↑𝑚 𝑦) ≼ ω))
5		oveq2 6984	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})))
6	5	breq1d 4939	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ↑𝑚 𝑥) ≼ ω ↔ (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≼ ω))
7		oveq2 6984	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 𝐵))
8	7	breq1d 4939	. 2 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑𝑚 𝑥) ≼ ω ↔ (𝐴 ↑𝑚 𝐵) ≼ ω))
9		mpct.a	. . . . 5 ⊢ (𝜑 → 𝐴 ≼ ω)
10		ctex 8321	. . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12		mapdm0 8221	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 ∅) = {∅})
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ↑𝑚 ∅) = {∅})
14		snfi 8391	. . . . 5 ⊢ {∅} ∈ Fin
15		fict 8910	. . . . 5 ⊢ ({∅} ∈ Fin → {∅} ≼ ω)
16	14, 15	ax-mp 5	. . . 4 ⊢ {∅} ≼ ω
17	16	a1i 11	. . 3 ⊢ (𝜑 → {∅} ≼ ω)
18	13, 17	eqbrtrd 4951	. 2 ⊢ (𝜑 → (𝐴 ↑𝑚 ∅) ≼ ω)
19		vex 3419	. . . . . 6 ⊢ 𝑦 ∈ V
20	19	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ≼ ω) → 𝑦 ∈ V)
21		snex 5188	. . . . . 6 ⊢ {𝑧} ∈ V
22	21	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ≼ ω) → {𝑧} ∈ V)
23	11	ad2antrr 713	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ≼ ω) → 𝐴 ∈ V)
24		eldifn 3995	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐵 ∖ 𝑦) → ¬ 𝑧 ∈ 𝑦)
25		disjsn 4521	. . . . . . . 8 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
26	24, 25	sylibr 226	. . . . . . 7 ⊢ (𝑧 ∈ (𝐵 ∖ 𝑦) → (𝑦 ∩ {𝑧}) = ∅)
27	26	adantl 474	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
28	27	ad2antlr 714	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ≼ ω) → (𝑦 ∩ {𝑧}) = ∅)
29		mapunen 8482	. . . . 5 ⊢ (((𝑦 ∈ V ∧ {𝑧} ∈ V ∧ 𝐴 ∈ V) ∧ (𝑦 ∩ {𝑧}) = ∅) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})))
30	20, 22, 23, 28, 29	syl31anc 1353	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ≼ ω) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})))
31		simpr 477	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ≼ ω) → (𝐴 ↑𝑚 𝑦) ≼ ω)
32		vex 3419	. . . . . . . . 9 ⊢ 𝑧 ∈ V
33	32	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑧 ∈ V)
34	11, 33	mapsnend 8385	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴)
35		endomtr 8364	. . . . . . 7 ⊢ (((𝐴 ↑𝑚 {𝑧}) ≈ 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 ↑𝑚 {𝑧}) ≼ ω)
36	34, 9, 35	syl2anc 576	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑𝑚 {𝑧}) ≼ ω)
37	36	ad2antrr 713	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ≼ ω) → (𝐴 ↑𝑚 {𝑧}) ≼ ω)
38		xpct 9236	. . . . 5 ⊢ (((𝐴 ↑𝑚 𝑦) ≼ ω ∧ (𝐴 ↑𝑚 {𝑧}) ≼ ω) → ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ≼ ω)
39	31, 37, 38	syl2anc 576	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ≼ ω) → ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ≼ ω)
40		endomtr 8364	. . . 4 ⊢ (((𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ∧ ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ≼ ω) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≼ ω)
41	30, 39, 40	syl2anc 576	. . 3 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ↑𝑚 𝑦) ≼ ω) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≼ ω)
42	41	ex 405	. 2 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) → ((𝐴 ↑𝑚 𝑦) ≼ ω → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≼ ω))
43		mpct.b	. 2 ⊢ (𝜑 → 𝐵 ∈ Fin)
44	2, 4, 6, 8, 18, 42, 43	findcard2d 8555	1 ⊢ (𝜑 → (𝐴 ↑𝑚 𝐵) ≼ ω)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  Vcvv 3416   ∖ cdif 3827   ∪ cun 3828   ∩ cin 3829   ⊆ wss 3830  ∅c0 4179  {csn 4441   class class class wbr 4929   × cxp 5405  (class class class)co 6976  ωcom 7396   ↑_𝑚 cmap 8206   ≈ cen 8303   ≼ cdom 8304  Fincfn 8306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-oi 8769  df-card 9162

This theorem is referenced by:  opnvonmbllem2  42344  smfmullem4  42498