MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunfi Structured version   Visualization version   GIF version

Theorem iunfi 8414
Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 8415. Note that 𝐵 depends on 𝑥, i.e. can be thought of as 𝐵(𝑥). (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
iunfi ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ Fin) → 𝑥𝐴 𝐵 ∈ Fin)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunfi
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3287 . . . 4 (𝑤 = ∅ → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ Fin))
2 iuneq1 4669 . . . . . 6 (𝑤 = ∅ → 𝑥𝑤 𝐵 = 𝑥 ∈ ∅ 𝐵)
3 0iun 4712 . . . . . 6 𝑥 ∈ ∅ 𝐵 = ∅
42, 3syl6eq 2821 . . . . 5 (𝑤 = ∅ → 𝑥𝑤 𝐵 = ∅)
54eleq1d 2835 . . . 4 (𝑤 = ∅ → ( 𝑥𝑤 𝐵 ∈ Fin ↔ ∅ ∈ Fin))
61, 5imbi12d 333 . . 3 (𝑤 = ∅ → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)))
7 raleq 3287 . . . 4 (𝑤 = 𝑦 → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥𝑦 𝐵 ∈ Fin))
8 iuneq1 4669 . . . . 5 (𝑤 = 𝑦 𝑥𝑤 𝐵 = 𝑥𝑦 𝐵)
98eleq1d 2835 . . . 4 (𝑤 = 𝑦 → ( 𝑥𝑤 𝐵 ∈ Fin ↔ 𝑥𝑦 𝐵 ∈ Fin))
107, 9imbi12d 333 . . 3 (𝑤 = 𝑦 → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥𝑦 𝐵 ∈ Fin → 𝑥𝑦 𝐵 ∈ Fin)))
11 raleq 3287 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
12 iuneq1 4669 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑥𝑤 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
1312eleq1d 2835 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ( 𝑥𝑤 𝐵 ∈ Fin ↔ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
1411, 13imbi12d 333 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)))
15 raleq 3287 . . . 4 (𝑤 = 𝐴 → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥𝐴 𝐵 ∈ Fin))
16 iuneq1 4669 . . . . 5 (𝑤 = 𝐴 𝑥𝑤 𝐵 = 𝑥𝐴 𝐵)
1716eleq1d 2835 . . . 4 (𝑤 = 𝐴 → ( 𝑥𝑤 𝐵 ∈ Fin ↔ 𝑥𝐴 𝐵 ∈ Fin))
1815, 17imbi12d 333 . . 3 (𝑤 = 𝐴 → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥𝐴 𝐵 ∈ Fin → 𝑥𝐴 𝐵 ∈ Fin)))
19 0fin 8348 . . . 4 ∅ ∈ Fin
2019a1i 11 . . 3 (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)
21 ssun1 3927 . . . . . . 7 𝑦 ⊆ (𝑦 ∪ {𝑧})
22 ssralv 3815 . . . . . . 7 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∀𝑥𝑦 𝐵 ∈ Fin))
2321, 22ax-mp 5 . . . . . 6 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∀𝑥𝑦 𝐵 ∈ Fin)
2423imim1i 63 . . . . 5 ((∀𝑥𝑦 𝐵 ∈ Fin → 𝑥𝑦 𝐵 ∈ Fin) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥𝑦 𝐵 ∈ Fin))
25 iunxun 4740 . . . . . . 7 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
26 nfcv 2913 . . . . . . . . . . 11 𝑦𝐵
27 nfcsb1v 3698 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵
28 csbeq1a 3691 . . . . . . . . . . 11 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
2926, 27, 28cbviun 4692 . . . . . . . . . 10 𝑥 ∈ {𝑧}𝐵 = 𝑦 ∈ {𝑧}𝑦 / 𝑥𝐵
30 vex 3354 . . . . . . . . . . 11 𝑧 ∈ V
31 csbeq1 3685 . . . . . . . . . . 11 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3230, 31iunxsn 4738 . . . . . . . . . 10 𝑦 ∈ {𝑧}𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
3329, 32eqtri 2793 . . . . . . . . 9 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵
34 ssun2 3928 . . . . . . . . . . 11 {𝑧} ⊆ (𝑦 ∪ {𝑧})
35 vsnid 4349 . . . . . . . . . . 11 𝑧 ∈ {𝑧}
3634, 35sselii 3749 . . . . . . . . . 10 𝑧 ∈ (𝑦 ∪ {𝑧})
37 nfcsb1v 3698 . . . . . . . . . . . 12 𝑥𝑧 / 𝑥𝐵
3837nfel1 2928 . . . . . . . . . . 11 𝑥𝑧 / 𝑥𝐵 ∈ Fin
39 csbeq1a 3691 . . . . . . . . . . . 12 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
4039eleq1d 2835 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐵 ∈ Fin ↔ 𝑧 / 𝑥𝐵 ∈ Fin))
4138, 40rspc 3454 . . . . . . . . . 10 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑧 / 𝑥𝐵 ∈ Fin))
4236, 41ax-mp 5 . . . . . . . . 9 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑧 / 𝑥𝐵 ∈ Fin)
4333, 42syl5eqel 2854 . . . . . . . 8 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥 ∈ {𝑧}𝐵 ∈ Fin)
44 unfi 8387 . . . . . . . 8 (( 𝑥𝑦 𝐵 ∈ Fin ∧ 𝑥 ∈ {𝑧}𝐵 ∈ Fin) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
4543, 44sylan2 580 . . . . . . 7 (( 𝑥𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
4625, 45syl5eqel 2854 . . . . . 6 (( 𝑥𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)
4746expcom 398 . . . . 5 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ( 𝑥𝑦 𝐵 ∈ Fin → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
4824, 47sylcom 30 . . . 4 ((∀𝑥𝑦 𝐵 ∈ Fin → 𝑥𝑦 𝐵 ∈ Fin) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
4948a1i 11 . . 3 (𝑦 ∈ Fin → ((∀𝑥𝑦 𝐵 ∈ Fin → 𝑥𝑦 𝐵 ∈ Fin) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)))
506, 10, 14, 18, 20, 49findcard2 8360 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴 𝐵 ∈ Fin → 𝑥𝐴 𝐵 ∈ Fin))
5150imp 393 1 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ Fin) → 𝑥𝐴 𝐵 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  csb 3682  cun 3721  wss 3723  c0 4063  {csn 4317   ciun 4655  Fincfn 8113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-en 8114  df-fin 8117
This theorem is referenced by:  unifi  8415  infssuni  8417  ixpfi  8423  ackbij1lem9  9256  ackbij1lem10  9257  fsuppmapnn0fiublem  12997  fsuppmapnn0fiub  12998  fsum2dlem  14709  fsumcom2  14713  fsumiun  14760  hashiun  14761  hash2iun  14762  ackbijnn  14767  fprod2dlem  14917  fprodcom2  14921  ablfaclem3  18694  pmatcoe1fsupp  20726  locfincmp  21550  txcmplem2  21666  alexsubALTlem3  22073  aannenlem1  24303  fsumvma  25159  numedglnl  26261  fsumiunle  29915  poimirlem30  33771  fiphp3d  37907  hbt  38224  cnrefiisplem  40568
  Copyright terms: Public domain W3C validator