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Theorem ssfirab 6568
Description: A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.)
Hypotheses
Ref Expression
ssfirab.a (𝜑𝐴 ∈ Fin)
ssfirab.dc (𝜑 → ∀𝑥𝐴 DECID 𝜓)
Assertion
Ref Expression
ssfirab (𝜑 → {𝑥𝐴𝜓} ∈ Fin)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ssfirab
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabeq 2604 . . 3 (𝑤 = ∅ → {𝑥𝑤𝜓} = {𝑥 ∈ ∅ ∣ 𝜓})
21eleq1d 2151 . 2 (𝑤 = ∅ → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin))
3 rabeq 2604 . . 3 (𝑤 = 𝑦 → {𝑥𝑤𝜓} = {𝑥𝑦𝜓})
43eleq1d 2151 . 2 (𝑤 = 𝑦 → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥𝑦𝜓} ∈ Fin))
5 rabeq 2604 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥𝑤𝜓} = {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓})
65eleq1d 2151 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin))
7 rabeq 2604 . . 3 (𝑤 = 𝐴 → {𝑥𝑤𝜓} = {𝑥𝐴𝜓})
87eleq1d 2151 . 2 (𝑤 = 𝐴 → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥𝐴𝜓} ∈ Fin))
9 rab0 3294 . . . 4 {𝑥 ∈ ∅ ∣ 𝜓} = ∅
10 0fin 6530 . . . 4 ∅ ∈ Fin
119, 10eqeltri 2155 . . 3 {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin
1211a1i 9 . 2 (𝜑 → {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin)
13 rabun2 3261 . . . . 5 {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓})
14 sbsbc 2830 . . . . . . . . . 10 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜓)
15 vex 2615 . . . . . . . . . . 11 𝑧 ∈ V
16 ralsns 3455 . . . . . . . . . . 11 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝜓[𝑧 / 𝑥]𝜓))
1715, 16ax-mp 7 . . . . . . . . . 10 (∀𝑥 ∈ {𝑧}𝜓[𝑧 / 𝑥]𝜓)
1814, 17bitr4i 185 . . . . . . . . 9 ([𝑧 / 𝑥]𝜓 ↔ ∀𝑥 ∈ {𝑧}𝜓)
19 rabid2 2536 . . . . . . . . 9 ({𝑧} = {𝑥 ∈ {𝑧} ∣ 𝜓} ↔ ∀𝑥 ∈ {𝑧}𝜓)
2018, 19sylbb2 136 . . . . . . . 8 ([𝑧 / 𝑥]𝜓 → {𝑧} = {𝑥 ∈ {𝑧} ∣ 𝜓})
2120adantl 271 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑧} = {𝑥 ∈ {𝑧} ∣ 𝜓})
2221uneq2d 3138 . . . . . 6 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑧}) = ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}))
23 simplr 497 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑥𝑦𝜓} ∈ Fin)
2415a1i 9 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → 𝑧 ∈ V)
25 simprr 499 . . . . . . . . . 10 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
2625ad2antrr 472 . . . . . . . . 9 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → 𝑧 ∈ (𝐴𝑦))
2726eldifbd 2996 . . . . . . . 8 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧𝑦)
28 elrabi 2756 . . . . . . . 8 (𝑧 ∈ {𝑥𝑦𝜓} → 𝑧𝑦)
2927, 28nsyl 591 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧 ∈ {𝑥𝑦𝜓})
30 unsnfi 6556 . . . . . . 7 (({𝑥𝑦𝜓} ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ {𝑥𝑦𝜓}) → ({𝑥𝑦𝜓} ∪ {𝑧}) ∈ Fin)
3123, 24, 29, 30syl3anc 1170 . . . . . 6 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑧}) ∈ Fin)
3222, 31eqeltrrd 2160 . . . . 5 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) ∈ Fin)
3313, 32syl5eqel 2169 . . . 4 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
34 ralsns 3455 . . . . . . . . . . . 12 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧} ¬ 𝜓[𝑧 / 𝑥] ¬ 𝜓))
3515, 34ax-mp 7 . . . . . . . . . . 11 (∀𝑥 ∈ {𝑧} ¬ 𝜓[𝑧 / 𝑥] ¬ 𝜓)
36 sbsbc 2830 . . . . . . . . . . 11 ([𝑧 / 𝑥] ¬ 𝜓[𝑧 / 𝑥] ¬ 𝜓)
37 sbn 1869 . . . . . . . . . . 11 ([𝑧 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑧 / 𝑥]𝜓)
3835, 36, 373bitr2ri 207 . . . . . . . . . 10 (¬ [𝑧 / 𝑥]𝜓 ↔ ∀𝑥 ∈ {𝑧} ¬ 𝜓)
39 rabeq0 3295 . . . . . . . . . 10 ({𝑥 ∈ {𝑧} ∣ 𝜓} = ∅ ↔ ∀𝑥 ∈ {𝑧} ¬ 𝜓)
4038, 39sylbb2 136 . . . . . . . . 9 (¬ [𝑧 / 𝑥]𝜓 → {𝑥 ∈ {𝑧} ∣ 𝜓} = ∅)
4140adantl 271 . . . . . . . 8 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ {𝑧} ∣ 𝜓} = ∅)
4241uneq2d 3138 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) = ({𝑥𝑦𝜓} ∪ ∅))
43 un0 3299 . . . . . . 7 ({𝑥𝑦𝜓} ∪ ∅) = {𝑥𝑦𝜓}
4442, 43syl6eq 2131 . . . . . 6 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) = {𝑥𝑦𝜓})
4513, 44syl5eq 2127 . . . . 5 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = {𝑥𝑦𝜓})
46 simplr 497 . . . . 5 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥𝑦𝜓} ∈ Fin)
4745, 46eqeltrd 2159 . . . 4 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
48 simplrr 503 . . . . . . 7 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → 𝑧 ∈ (𝐴𝑦))
4948eldifad 2995 . . . . . 6 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → 𝑧𝐴)
50 ssfirab.dc . . . . . . 7 (𝜑 → ∀𝑥𝐴 DECID 𝜓)
5150ad3antrrr 476 . . . . . 6 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → ∀𝑥𝐴 DECID 𝜓)
52 nfs1v 1858 . . . . . . . 8 𝑥[𝑧 / 𝑥]𝜓
5352nfdc 1590 . . . . . . 7 𝑥DECID [𝑧 / 𝑥]𝜓
54 sbequ12 1696 . . . . . . . 8 (𝑥 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
5554dcbid 782 . . . . . . 7 (𝑥 = 𝑧 → (DECID 𝜓DECID [𝑧 / 𝑥]𝜓))
5653, 55rspc 2706 . . . . . 6 (𝑧𝐴 → (∀𝑥𝐴 DECID 𝜓DECID [𝑧 / 𝑥]𝜓))
5749, 51, 56sylc 61 . . . . 5 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → DECID [𝑧 / 𝑥]𝜓)
58 exmiddc 778 . . . . 5 (DECID [𝑧 / 𝑥]𝜓 → ([𝑧 / 𝑥]𝜓 ∨ ¬ [𝑧 / 𝑥]𝜓))
5957, 58syl 14 . . . 4 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → ([𝑧 / 𝑥]𝜓 ∨ ¬ [𝑧 / 𝑥]𝜓))
6033, 47, 59mpjaodan 745 . . 3 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
6160ex 113 . 2 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ({𝑥𝑦𝜓} ∈ Fin → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin))
62 ssfirab.a . 2 (𝜑𝐴 ∈ Fin)
632, 4, 6, 8, 12, 61, 62findcard2sd 6538 1 (𝜑 → {𝑥𝐴𝜓} ∈ Fin)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 776   = wceq 1285  wcel 1434  [wsb 1687  wral 2353  {crab 2357  Vcvv 2612  [wsbc 2826  cdif 2981  cun 2982  wss 2984  c0 3269  {csn 3422  Fincfn 6387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1o 6113  df-er 6222  df-en 6388  df-fin 6390
This theorem is referenced by:  ssfidc  6569  phivalfi  10968  hashdvds  10977  phiprmpw  10978  phimullem  10981  hashgcdeq  10984
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