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Theorem iunfi 9385
Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 9386. 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 3312 . . . 4 (𝑤 = ∅ → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ Fin))
2 iuneq1 5017 . . . . . 6 (𝑤 = ∅ → 𝑥𝑤 𝐵 = 𝑥 ∈ ∅ 𝐵)
3 0iun 5071 . . . . . 6 𝑥 ∈ ∅ 𝐵 = ∅
42, 3eqtrdi 2782 . . . . 5 (𝑤 = ∅ → 𝑥𝑤 𝐵 = ∅)
54eleq1d 2811 . . . 4 (𝑤 = ∅ → ( 𝑥𝑤 𝐵 ∈ Fin ↔ ∅ ∈ Fin))
61, 5imbi12d 343 . . 3 (𝑤 = ∅ → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)))
7 raleq 3312 . . . 4 (𝑤 = 𝑦 → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥𝑦 𝐵 ∈ Fin))
8 iuneq1 5017 . . . . 5 (𝑤 = 𝑦 𝑥𝑤 𝐵 = 𝑥𝑦 𝐵)
98eleq1d 2811 . . . 4 (𝑤 = 𝑦 → ( 𝑥𝑤 𝐵 ∈ Fin ↔ 𝑥𝑦 𝐵 ∈ Fin))
107, 9imbi12d 343 . . 3 (𝑤 = 𝑦 → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥𝑦 𝐵 ∈ Fin → 𝑥𝑦 𝐵 ∈ Fin)))
11 raleq 3312 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
12 iuneq1 5017 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑥𝑤 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
1312eleq1d 2811 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ( 𝑥𝑤 𝐵 ∈ Fin ↔ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
1411, 13imbi12d 343 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)))
15 raleq 3312 . . . 4 (𝑤 = 𝐴 → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥𝐴 𝐵 ∈ Fin))
16 iuneq1 5017 . . . . 5 (𝑤 = 𝐴 𝑥𝑤 𝐵 = 𝑥𝐴 𝐵)
1716eleq1d 2811 . . . 4 (𝑤 = 𝐴 → ( 𝑥𝑤 𝐵 ∈ Fin ↔ 𝑥𝐴 𝐵 ∈ Fin))
1815, 17imbi12d 343 . . 3 (𝑤 = 𝐴 → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥𝐴 𝐵 ∈ Fin → 𝑥𝐴 𝐵 ∈ Fin)))
19 0fi 9080 . . . 4 ∅ ∈ Fin
2019a1i 11 . . 3 (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)
21 ssun1 4173 . . . . . . 7 𝑦 ⊆ (𝑦 ∪ {𝑧})
22 ssralv 4048 . . . . . . 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 5102 . . . . . . 7 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
26 nfcv 2892 . . . . . . . . . . 11 𝑦𝐵
27 nfcsb1v 3917 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵
28 csbeq1a 3906 . . . . . . . . . . 11 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
2926, 27, 28cbviun 5044 . . . . . . . . . 10 𝑥 ∈ {𝑧}𝐵 = 𝑦 ∈ {𝑧}𝑦 / 𝑥𝐵
30 vex 3466 . . . . . . . . . . 11 𝑧 ∈ V
31 csbeq1 3895 . . . . . . . . . . 11 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3230, 31iunxsn 5099 . . . . . . . . . 10 𝑦 ∈ {𝑧}𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
3329, 32eqtri 2754 . . . . . . . . 9 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵
34 ssun2 4174 . . . . . . . . . . 11 {𝑧} ⊆ (𝑦 ∪ {𝑧})
35 vsnid 4670 . . . . . . . . . . 11 𝑧 ∈ {𝑧}
3634, 35sselii 3976 . . . . . . . . . 10 𝑧 ∈ (𝑦 ∪ {𝑧})
37 nfcsb1v 3917 . . . . . . . . . . . 12 𝑥𝑧 / 𝑥𝐵
3837nfel1 2909 . . . . . . . . . . 11 𝑥𝑧 / 𝑥𝐵 ∈ Fin
39 csbeq1a 3906 . . . . . . . . . . . 12 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
4039eleq1d 2811 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐵 ∈ Fin ↔ 𝑧 / 𝑥𝐵 ∈ Fin))
4138, 40rspc 3596 . . . . . . . . . 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, 42eqeltrid 2830 . . . . . . . 8 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥 ∈ {𝑧}𝐵 ∈ Fin)
44 unfi 9210 . . . . . . . 8 (( 𝑥𝑦 𝐵 ∈ Fin ∧ 𝑥 ∈ {𝑧}𝐵 ∈ Fin) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
4543, 44sylan2 591 . . . . . . 7 (( 𝑥𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
4625, 45eqeltrid 2830 . . . . . 6 (( 𝑥𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)
4746expcom 412 . . . . 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 9202 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴 𝐵 ∈ Fin → 𝑥𝐴 𝐵 ∈ Fin))
5150imp 405 1 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ Fin) → 𝑥𝐴 𝐵 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wral 3051  csb 3892  cun 3945  wss 3947  c0 4325  {csn 4633   ciun 5001  Fincfn 8974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-om 7877  df-en 8975  df-fin 8978
This theorem is referenced by:  unifi  9386  infssuni  9388  ixpfi  9393  ackbij1lem9  10271  ackbij1lem10  10272  fsuppmapnn0fiublem  14010  fsuppmapnn0fiub  14011  fsum2dlem  15774  fsumcom2  15778  fsumiun  15825  hashiun  15826  hash2iun  15827  ackbijnn  15832  fprod2dlem  15982  fprodcom2  15986  ablfaclem3  20087  pmatcoe1fsupp  22694  locfincmp  23521  txcmplem2  23637  alexsubALTlem3  24044  aannenlem1  26356  fsumvma  27242  numedglnl  29080  fsumiunle  32730  fedgmullem1  33524  poimirlem30  37351  fiphp3d  42476  hbt  42791  cnrefiisplem  45450
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