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Theorem iunfi 8607
Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 8608. 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 3346 . . . 4 (𝑤 = ∅ → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ Fin))
2 iuneq1 4807 . . . . . 6 (𝑤 = ∅ → 𝑥𝑤 𝐵 = 𝑥 ∈ ∅ 𝐵)
3 0iun 4852 . . . . . 6 𝑥 ∈ ∅ 𝐵 = ∅
42, 3syl6eq 2831 . . . . 5 (𝑤 = ∅ → 𝑥𝑤 𝐵 = ∅)
54eleq1d 2851 . . . 4 (𝑤 = ∅ → ( 𝑥𝑤 𝐵 ∈ Fin ↔ ∅ ∈ Fin))
61, 5imbi12d 337 . . 3 (𝑤 = ∅ → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)))
7 raleq 3346 . . . 4 (𝑤 = 𝑦 → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥𝑦 𝐵 ∈ Fin))
8 iuneq1 4807 . . . . 5 (𝑤 = 𝑦 𝑥𝑤 𝐵 = 𝑥𝑦 𝐵)
98eleq1d 2851 . . . 4 (𝑤 = 𝑦 → ( 𝑥𝑤 𝐵 ∈ Fin ↔ 𝑥𝑦 𝐵 ∈ Fin))
107, 9imbi12d 337 . . 3 (𝑤 = 𝑦 → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥𝑦 𝐵 ∈ Fin → 𝑥𝑦 𝐵 ∈ Fin)))
11 raleq 3346 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
12 iuneq1 4807 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑥𝑤 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
1312eleq1d 2851 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ( 𝑥𝑤 𝐵 ∈ Fin ↔ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
1411, 13imbi12d 337 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)))
15 raleq 3346 . . . 4 (𝑤 = 𝐴 → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥𝐴 𝐵 ∈ Fin))
16 iuneq1 4807 . . . . 5 (𝑤 = 𝐴 𝑥𝑤 𝐵 = 𝑥𝐴 𝐵)
1716eleq1d 2851 . . . 4 (𝑤 = 𝐴 → ( 𝑥𝑤 𝐵 ∈ Fin ↔ 𝑥𝐴 𝐵 ∈ Fin))
1815, 17imbi12d 337 . . 3 (𝑤 = 𝐴 → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥𝐴 𝐵 ∈ Fin → 𝑥𝐴 𝐵 ∈ Fin)))
19 0fin 8541 . . . 4 ∅ ∈ Fin
2019a1i 11 . . 3 (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)
21 ssun1 4038 . . . . . . 7 𝑦 ⊆ (𝑦 ∪ {𝑧})
22 ssralv 3924 . . . . . . 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 4882 . . . . . . 7 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
26 nfcv 2933 . . . . . . . . . . 11 𝑦𝐵
27 nfcsb1v 3805 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵
28 csbeq1a 3796 . . . . . . . . . . 11 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
2926, 27, 28cbviun 4831 . . . . . . . . . 10 𝑥 ∈ {𝑧}𝐵 = 𝑦 ∈ {𝑧}𝑦 / 𝑥𝐵
30 vex 3419 . . . . . . . . . . 11 𝑧 ∈ V
31 csbeq1 3790 . . . . . . . . . . 11 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3230, 31iunxsn 4879 . . . . . . . . . 10 𝑦 ∈ {𝑧}𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
3329, 32eqtri 2803 . . . . . . . . 9 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵
34 ssun2 4039 . . . . . . . . . . 11 {𝑧} ⊆ (𝑦 ∪ {𝑧})
35 vsnid 4474 . . . . . . . . . . 11 𝑧 ∈ {𝑧}
3634, 35sselii 3856 . . . . . . . . . 10 𝑧 ∈ (𝑦 ∪ {𝑧})
37 nfcsb1v 3805 . . . . . . . . . . . 12 𝑥𝑧 / 𝑥𝐵
3837nfel1 2947 . . . . . . . . . . 11 𝑥𝑧 / 𝑥𝐵 ∈ Fin
39 csbeq1a 3796 . . . . . . . . . . . 12 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
4039eleq1d 2851 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐵 ∈ Fin ↔ 𝑧 / 𝑥𝐵 ∈ Fin))
4138, 40rspc 3530 . . . . . . . . . 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 2871 . . . . . . . 8 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥 ∈ {𝑧}𝐵 ∈ Fin)
44 unfi 8580 . . . . . . . 8 (( 𝑥𝑦 𝐵 ∈ Fin ∧ 𝑥 ∈ {𝑧}𝐵 ∈ Fin) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
4543, 44sylan2 583 . . . . . . 7 (( 𝑥𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
4625, 45syl5eqel 2871 . . . . . 6 (( 𝑥𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)
4746expcom 406 . . . . 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 8553 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴 𝐵 ∈ Fin → 𝑥𝐴 𝐵 ∈ Fin))
5150imp 398 1 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ Fin) → 𝑥𝐴 𝐵 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  wral 3089  csb 3787  cun 3828  wss 3830  c0 4179  {csn 4441   ciun 4792  Fincfn 8306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-en 8307  df-fin 8310
This theorem is referenced by:  unifi  8608  infssuni  8610  ixpfi  8616  ackbij1lem9  9448  ackbij1lem10  9449  fsuppmapnn0fiublem  13173  fsuppmapnn0fiub  13174  fsum2dlem  14985  fsumcom2  14989  fsumiun  15036  hashiun  15037  hash2iun  15038  ackbijnn  15043  fprod2dlem  15194  fprodcom2  15198  ablfaclem3  18959  pmatcoe1fsupp  21013  locfincmp  21838  txcmplem2  21954  alexsubALTlem3  22361  aannenlem1  24620  fsumvma  25491  numedglnl  26632  fsumiunle  30291  fedgmullem1  30651  poimirlem30  34360  fiphp3d  38809  hbt  39123  cnrefiisplem  41539
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