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Theorem iunfi 8409
Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 8410. 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 3286 . . . 4 (𝑤 = ∅ → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ Fin))
2 iuneq1 4666 . . . . . 6 (𝑤 = ∅ → 𝑥𝑤 𝐵 = 𝑥 ∈ ∅ 𝐵)
3 0iun 4709 . . . . . 6 𝑥 ∈ ∅ 𝐵 = ∅
42, 3syl6eq 2820 . . . . 5 (𝑤 = ∅ → 𝑥𝑤 𝐵 = ∅)
54eleq1d 2834 . . . 4 (𝑤 = ∅ → ( 𝑥𝑤 𝐵 ∈ Fin ↔ ∅ ∈ Fin))
61, 5imbi12d 333 . . 3 (𝑤 = ∅ → ((∀𝑥𝑤 𝐵 ∈ Fin → 𝑥𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)))
7 raleq 3286 . . . 4 (𝑤 = 𝑦 → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥𝑦 𝐵 ∈ Fin))
8 iuneq1 4666 . . . . 5 (𝑤 = 𝑦 𝑥𝑤 𝐵 = 𝑥𝑦 𝐵)
98eleq1d 2834 . . . 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 3286 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
12 iuneq1 4666 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑥𝑤 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
1312eleq1d 2834 . . . 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 3286 . . . 4 (𝑤 = 𝐴 → (∀𝑥𝑤 𝐵 ∈ Fin ↔ ∀𝑥𝐴 𝐵 ∈ Fin))
16 iuneq1 4666 . . . . 5 (𝑤 = 𝐴 𝑥𝑤 𝐵 = 𝑥𝐴 𝐵)
1716eleq1d 2834 . . . 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 8343 . . . 4 ∅ ∈ Fin
2019a1i 11 . . 3 (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)
21 ssun1 3925 . . . . . . 7 𝑦 ⊆ (𝑦 ∪ {𝑧})
22 ssralv 3813 . . . . . . 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 4737 . . . . . . 7 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
26 nfcv 2912 . . . . . . . . . . 11 𝑦𝐵
27 nfcsb1v 3696 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵
28 csbeq1a 3689 . . . . . . . . . . 11 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
2926, 27, 28cbviun 4689 . . . . . . . . . 10 𝑥 ∈ {𝑧}𝐵 = 𝑦 ∈ {𝑧}𝑦 / 𝑥𝐵
30 vex 3352 . . . . . . . . . . 11 𝑧 ∈ V
31 csbeq1 3683 . . . . . . . . . . 11 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3230, 31iunxsn 4735 . . . . . . . . . 10 𝑦 ∈ {𝑧}𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵
3329, 32eqtri 2792 . . . . . . . . 9 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵
34 ssun2 3926 . . . . . . . . . . 11 {𝑧} ⊆ (𝑦 ∪ {𝑧})
35 vsnid 4346 . . . . . . . . . . 11 𝑧 ∈ {𝑧}
3634, 35sselii 3747 . . . . . . . . . 10 𝑧 ∈ (𝑦 ∪ {𝑧})
37 nfcsb1v 3696 . . . . . . . . . . . 12 𝑥𝑧 / 𝑥𝐵
3837nfel1 2927 . . . . . . . . . . 11 𝑥𝑧 / 𝑥𝐵 ∈ Fin
39 csbeq1a 3689 . . . . . . . . . . . 12 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
4039eleq1d 2834 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝐵 ∈ Fin ↔ 𝑧 / 𝑥𝐵 ∈ Fin))
4138, 40rspc 3452 . . . . . . . . . 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 2853 . . . . . . . 8 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → 𝑥 ∈ {𝑧}𝐵 ∈ Fin)
44 unfi 8382 . . . . . . . 8 (( 𝑥𝑦 𝐵 ∈ Fin ∧ 𝑥 ∈ {𝑧}𝐵 ∈ Fin) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
4543, 44sylan2 572 . . . . . . 7 (( 𝑥𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
4625, 45syl5eqel 2853 . . . . . 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 8355 . 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 1630  wcel 2144  wral 3060  csb 3680  cun 3719  wss 3721  c0 4061  {csn 4314   ciun 4652  Fincfn 8108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-fin 8112
This theorem is referenced by:  unifi  8410  infssuni  8412  ixpfi  8418  ackbij1lem9  9251  ackbij1lem10  9252  fsuppmapnn0fiublem  12996  fsuppmapnn0fiub  12997  fsum2dlem  14708  fsumcom2  14712  fsumiun  14759  hashiun  14760  hash2iun  14761  ackbijnn  14766  fprod2dlem  14916  fprodcom2  14920  ablfaclem3  18693  pmatcoe1fsupp  20725  locfincmp  21549  txcmplem2  21665  alexsubALTlem3  22072  aannenlem1  24302  fsumvma  25158  numedglnl  26260  fsumiunle  29909  poimirlem30  33765  fiphp3d  37902  hbt  38219  cnrefiisplem  40567
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