Theorem iunfidisj 7213

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 4004	. . 3 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
2	1	eleq1d 2301	. 2 ⊢ (𝑤 = ∅ → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ ∅ 𝐵 ∈ Fin))
3		iuneq1 4004	. . 3 ⊢ (𝑤 = 𝑦 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵)
4	3	eleq1d 2301	. 2 ⊢ (𝑤 = 𝑦 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin))
5		iuneq1 4004	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
6	5	eleq1d 2301	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
7		iuneq1 4004	. . 3 ⊢ (𝑤 = 𝐴 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
8	7	eleq1d 2301	. 2 ⊢ (𝑤 = 𝐴 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin))
9		0iun 4049	. . . 4 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
10		0fi 7141	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2305	. . 3 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 ∈ Fin
12	11	a1i 9	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ ∅ 𝐵 ∈ Fin)
13		iunxun 4071	. . . 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 3171	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
16		csbeq1a 3147	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
17	15, 16	iunxsngf 4069	. . . . . . 7 ⊢ (𝑧 ∈ V → ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
18	17	elv 2817	. . . . . 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 3222	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → 𝑧 ∈ 𝐴)
21		simpll2 1064	. . . . . . 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 2395	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ Fin
23	16	eleq1d 2301	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ Fin ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ Fin))
24	22, 23	rspc 2915	. . . . . . 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 2319	. . . . 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 1065	. . . . . 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 3839	. . . . . 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 3223	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
31		disjsn 3751	. . . . . . 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 4104	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑦 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴 ∧ (𝑦 ∩ {𝑧}) = ∅)) → (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝑥 ∈ {𝑧}𝐵) = ∅)
34	27, 28, 29, 32, 33	syl13anc 1276	. . . . 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 7182	. . . . 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 1274	. . . 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 2319	. . 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 1024	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → 𝐴 ∈ Fin)
40	2, 4, 6, 8, 12, 38, 39	findcard2d 7148	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∀wral 2520  Vcvv 2813  ⦋csb 3138   ∖ cdif 3208   ∪ cun 3209   ∩ cin 3210   ⊆ wss 3211  ∅c0 3508  {csn 3689  ∪ ciun 3991  Disj wdisj 4085  Fincfn 6975

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-disj 4086  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1o 6647  df-er 6767  df-en 6976  df-fin 6978

This theorem is referenced by:  fsum2dlemstep  12120  fisumcom2  12124  fsumiun  12163  hashiun  12164  hash2iun  12165  fprod2dlemstep  12308  fprodcom2fi  12312