Theorem xpfiOLD 9331

Description: Obsolete version of xpfi 9330 as of 10-Jan-2025. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
xpfiOLD	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)

Proof of Theorem xpfiOLD

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

Step	Hyp	Ref	Expression
1		xpeq1 5668	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	eleq1d 2819	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 × 𝐵) ∈ Fin ↔ (∅ × 𝐵) ∈ Fin))
3	2	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)))
4		xpeq1 5668	. . . . 5 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∖ {𝑧}) × 𝐵))
5	4	eleq1d 2819	. . . 4 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥 × 𝐵) ∈ Fin ↔ ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin))
6	5	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin)))
7		xpeq1 5668	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
8	7	eleq1d 2819	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 × 𝐵) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
9	8	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)))
10		xpeq1 5668	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
11	10	eleq1d 2819	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 × 𝐵) ∈ Fin ↔ (𝐴 × 𝐵) ∈ Fin))
12	11	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (𝐴 × 𝐵) ∈ Fin)))
13		0xp 5753	. . . . 5 ⊢ (∅ × 𝐵) = ∅
14		0fi 9056	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2830	. . . 4 ⊢ (∅ × 𝐵) ∈ Fin
16	15	a1i 11	. . 3 ⊢ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)
17		neq0 4327	. . . . . . 7 ⊢ (¬ 𝑦 = ∅ ↔ ∃𝑤 𝑤 ∈ 𝑦)
18		sneq 4611	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
19	18	difeq2d 4101	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑦 ∖ {𝑧}) = (𝑦 ∖ {𝑤}))
20	19	xpeq1d 5683	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → ((𝑦 ∖ {𝑧}) × 𝐵) = ((𝑦 ∖ {𝑤}) × 𝐵))
21	20	eleq1d 2819	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin ↔ ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
22	21	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
23	22	rspcv 3597	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
24	23	adantl 481	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
25		pm2.27 42	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin) → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
26	25	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin) → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
27		snex 5406	. . . . . . . . . . . . . . 15 ⊢ {𝑤} ∈ V
28		xpexg 7744	. . . . . . . . . . . . . . 15 ⊢ (({𝑤} ∈ V ∧ 𝐵 ∈ Fin) → ({𝑤} × 𝐵) ∈ V)
29	27, 28	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ∈ V)
30		id 22	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ Fin)
31		vex 3463	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
32		2ndconst 8100	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ V → (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵)
33	31, 32	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Fin → (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵)
34		f1oen2g 8983	. . . . . . . . . . . . . 14 ⊢ ((({𝑤} × 𝐵) ∈ V ∧ 𝐵 ∈ Fin ∧ (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵) → ({𝑤} × 𝐵) ≈ 𝐵)
35	29, 30, 33, 34	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ≈ 𝐵)
36		enfii 9200	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ Fin ∧ ({𝑤} × 𝐵) ≈ 𝐵) → ({𝑤} × 𝐵) ∈ Fin)
37	35, 36	mpdan 687	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ∈ Fin)
38	37	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ({𝑤} × 𝐵) ∈ Fin)
39		unfi 9185	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
40		xpundir 5724	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵))
41		difsnid 4786	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑦 → ((𝑦 ∖ {𝑤}) ∪ {𝑤}) = 𝑦)
42	41	xpeq1d 5683	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ 𝑦 → (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (𝑦 × 𝐵))
43	40, 42	eqtr3id 2784	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝑦 → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) = (𝑦 × 𝐵))
44	43	eleq1d 2819	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝑦 → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
45	44	biimpd 229	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝑦 → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin → (𝑦 × 𝐵) ∈ Fin))
46	45	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin → (𝑦 × 𝐵) ∈ Fin))
47	39, 46	syl5 34	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
48	38, 47	mpan2d 694	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin → (𝑦 × 𝐵) ∈ Fin))
49	24, 26, 48	3syld 60	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
50	49	ex 412	. . . . . . . 8 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
51	50	exlimdv 1933	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑤 𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
52	17, 51	biimtrid 242	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (¬ 𝑦 = ∅ → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
53		xpeq1 5668	. . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 × 𝐵) = (∅ × 𝐵))
54	53, 15	eqeltrdi 2842	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 × 𝐵) ∈ Fin)
55	54	a1d 25	. . . . . 6 ⊢ (𝑦 = ∅ → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
56	52, 55	pm2.61d2 181	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
57	56	ex 412	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐵 ∈ Fin → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
58	57	com23 86	. . 3 ⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)))
59	3, 6, 9, 12, 16, 58	findcard 9177	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ∈ Fin → (𝐴 × 𝐵) ∈ Fin))
60	59	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3051  Vcvv 3459   ∖ cdif 3923   ∪ cun 3924  ∅c0 4308  {csn 4601   class class class wbr 5119   × cxp 5652   ↾ cres 5656  –1-1-onto→wf1o 6530  2^nd c2nd 7987   ≈ cen 8956  Fincfn 8959

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7862  df-1st 7988  df-2nd 7989  df-1o 8480  df-en 8960  df-fin 8963

This theorem is referenced by: (None)