Theorem iunfidisj 7144

Description: The finite union of disjoint finite sets is finite. Note that 𝐵 depends on 𝑥, i.e. can be thought of as 𝐵(𝑥). (Contributed by NM, 23-Mar-2006.) (Revised by Jim Kingdon, 7-Oct-2022.)

Assertion

Ref	Expression
iunfidisj	⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunfidisj

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

Step	Hyp	Ref	Expression
1		iuneq1 3983	. . 3 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
2	1	eleq1d 2300	. 2 ⊢ (𝑤 = ∅ → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ ∅ 𝐵 ∈ Fin))
3		iuneq1 3983	. . 3 ⊢ (𝑤 = 𝑦 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵)
4	3	eleq1d 2300	. 2 ⊢ (𝑤 = 𝑦 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin))
5		iuneq1 3983	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
6	5	eleq1d 2300	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
7		iuneq1 3983	. . 3 ⊢ (𝑤 = 𝐴 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
8	7	eleq1d 2300	. 2 ⊢ (𝑤 = 𝐴 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin))
9		0iun 4028	. . . 4 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
10		0fi 7072	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2304	. . 3 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 ∈ Fin
12	11	a1i 9	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ ∅ 𝐵 ∈ Fin)
13		iunxun 4050	. . . 4 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
14		simpr 110	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin)
15		nfcsb1v 3160	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
16		csbeq1a 3136	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
17	15, 16	iunxsngf 4048	. . . . . . 7 ⊢ (𝑧 ∈ V → ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
18	17	elv 2806	. . . . . 6 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
19		simplrr 538	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → 𝑧 ∈ (𝐴 ∖ 𝑦))
20	19	eldifad 3211	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → 𝑧 ∈ 𝐴)
21		simpll2 1063	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin)
22	15	nfel1 2385	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ Fin
23	16	eleq1d 2300	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ Fin ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ Fin))
24	22, 23	rspc 2904	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑧 / 𝑥⦌𝐵 ∈ Fin))
25	20, 21, 24	sylc 62	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → ⦋𝑧 / 𝑥⦌𝐵 ∈ Fin)
26	18, 25	eqeltrid 2318	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → ∪ 𝑥 ∈ {𝑧}𝐵 ∈ Fin)
27		simpll3 1064	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → Disj 𝑥 ∈ 𝐴 𝐵)
28		simplrl 537	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → 𝑦 ⊆ 𝐴)
29	20	snssd 3818	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → {𝑧} ⊆ 𝐴)
30	19	eldifbd 3212	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
31		disjsn 3731	. . . . . . 7 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
32	30, 31	sylibr 134	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → (𝑦 ∩ {𝑧}) = ∅)
33		disjiun 4083	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑦 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴 ∧ (𝑦 ∩ {𝑧}) = ∅)) → (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝑥 ∈ {𝑧}𝐵) = ∅)
34	27, 28, 29, 32, 33	syl13anc 1275	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝑥 ∈ {𝑧}𝐵) = ∅)
35		unfidisj 7113	. . . . 5 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ Fin ∧ (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝑥 ∈ {𝑧}𝐵) = ∅) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
36	14, 26, 34, 35	syl3anc 1273	. . . 4 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
37	13, 36	eqeltrid 2318	. . 3 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)
38	37	ex 115	. 2 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
39		simp1 1023	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → 𝐴 ∈ Fin)
40	2, 4, 6, 8, 12, 38, 39	findcard2d 7079	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802  ⦋csb 3127   ∖ cdif 3197   ∪ cun 3198   ∩ cin 3199   ⊆ wss 3200  ∅c0 3494  {csn 3669  ∪ ciun 3970  Disj wdisj 4064  Fincfn 6908

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-er 6701  df-en 6909  df-fin 6911

This theorem is referenced by:  fsum2dlemstep  11994  fisumcom2  11998  fsumiun  12037  hashiun  12038  hash2iun  12039  fprod2dlemstep  12182  fprodcom2fi  12186