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Theorem ssfirab 6790
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 2652 . . 3 (𝑤 = ∅ → {𝑥𝑤𝜓} = {𝑥 ∈ ∅ ∣ 𝜓})
21eleq1d 2186 . 2 (𝑤 = ∅ → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin))
3 rabeq 2652 . . 3 (𝑤 = 𝑦 → {𝑥𝑤𝜓} = {𝑥𝑦𝜓})
43eleq1d 2186 . 2 (𝑤 = 𝑦 → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥𝑦𝜓} ∈ Fin))
5 rabeq 2652 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥𝑤𝜓} = {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓})
65eleq1d 2186 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin))
7 rabeq 2652 . . 3 (𝑤 = 𝐴 → {𝑥𝑤𝜓} = {𝑥𝐴𝜓})
87eleq1d 2186 . 2 (𝑤 = 𝐴 → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥𝐴𝜓} ∈ Fin))
9 rab0 3361 . . . 4 {𝑥 ∈ ∅ ∣ 𝜓} = ∅
10 0fin 6746 . . . 4 ∅ ∈ Fin
119, 10eqeltri 2190 . . 3 {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin
1211a1i 9 . 2 (𝜑 → {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin)
13 rabun2 3325 . . . . 5 {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓})
14 sbsbc 2886 . . . . . . . . . 10 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜓)
15 vex 2663 . . . . . . . . . . 11 𝑧 ∈ V
16 ralsns 3532 . . . . . . . . . . 11 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝜓[𝑧 / 𝑥]𝜓))
1715, 16ax-mp 5 . . . . . . . . . 10 (∀𝑥 ∈ {𝑧}𝜓[𝑧 / 𝑥]𝜓)
1814, 17bitr4i 186 . . . . . . . . 9 ([𝑧 / 𝑥]𝜓 ↔ ∀𝑥 ∈ {𝑧}𝜓)
19 rabid2 2584 . . . . . . . . 9 ({𝑧} = {𝑥 ∈ {𝑧} ∣ 𝜓} ↔ ∀𝑥 ∈ {𝑧}𝜓)
2018, 19sylbb2 137 . . . . . . . 8 ([𝑧 / 𝑥]𝜓 → {𝑧} = {𝑥 ∈ {𝑧} ∣ 𝜓})
2120adantl 275 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑧} = {𝑥 ∈ {𝑧} ∣ 𝜓})
2221uneq2d 3200 . . . . . 6 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑧}) = ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}))
23 simplr 504 . . . . . . 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 506 . . . . . . . . . 10 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
2625ad2antrr 479 . . . . . . . . 9 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → 𝑧 ∈ (𝐴𝑦))
2726eldifbd 3053 . . . . . . . 8 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧𝑦)
28 elrabi 2810 . . . . . . . 8 (𝑧 ∈ {𝑥𝑦𝜓} → 𝑧𝑦)
2927, 28nsyl 602 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧 ∈ {𝑥𝑦𝜓})
30 unsnfi 6775 . . . . . . 7 (({𝑥𝑦𝜓} ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ {𝑥𝑦𝜓}) → ({𝑥𝑦𝜓} ∪ {𝑧}) ∈ Fin)
3123, 24, 29, 30syl3anc 1201 . . . . . 6 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑧}) ∈ Fin)
3222, 31eqeltrrd 2195 . . . . 5 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) ∈ Fin)
3313, 32eqeltrid 2204 . . . 4 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
34 ralsns 3532 . . . . . . . . . . . 12 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧} ¬ 𝜓[𝑧 / 𝑥] ¬ 𝜓))
3515, 34ax-mp 5 . . . . . . . . . . 11 (∀𝑥 ∈ {𝑧} ¬ 𝜓[𝑧 / 𝑥] ¬ 𝜓)
36 sbsbc 2886 . . . . . . . . . . 11 ([𝑧 / 𝑥] ¬ 𝜓[𝑧 / 𝑥] ¬ 𝜓)
37 sbn 1903 . . . . . . . . . . 11 ([𝑧 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑧 / 𝑥]𝜓)
3835, 36, 373bitr2ri 208 . . . . . . . . . 10 (¬ [𝑧 / 𝑥]𝜓 ↔ ∀𝑥 ∈ {𝑧} ¬ 𝜓)
39 rabeq0 3362 . . . . . . . . . 10 ({𝑥 ∈ {𝑧} ∣ 𝜓} = ∅ ↔ ∀𝑥 ∈ {𝑧} ¬ 𝜓)
4038, 39sylbb2 137 . . . . . . . . 9 (¬ [𝑧 / 𝑥]𝜓 → {𝑥 ∈ {𝑧} ∣ 𝜓} = ∅)
4140adantl 275 . . . . . . . 8 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ {𝑧} ∣ 𝜓} = ∅)
4241uneq2d 3200 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) = ({𝑥𝑦𝜓} ∪ ∅))
43 un0 3366 . . . . . . 7 ({𝑥𝑦𝜓} ∪ ∅) = {𝑥𝑦𝜓}
4442, 43syl6eq 2166 . . . . . 6 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) = {𝑥𝑦𝜓})
4513, 44syl5eq 2162 . . . . 5 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = {𝑥𝑦𝜓})
46 simplr 504 . . . . 5 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥𝑦𝜓} ∈ Fin)
4745, 46eqeltrd 2194 . . . 4 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
48 simplrr 510 . . . . . . 7 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → 𝑧 ∈ (𝐴𝑦))
4948eldifad 3052 . . . . . 6 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → 𝑧𝐴)
50 ssfirab.dc . . . . . . 7 (𝜑 → ∀𝑥𝐴 DECID 𝜓)
5150ad3antrrr 483 . . . . . 6 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → ∀𝑥𝐴 DECID 𝜓)
52 nfs1v 1892 . . . . . . . 8 𝑥[𝑧 / 𝑥]𝜓
5352nfdc 1622 . . . . . . 7 𝑥DECID [𝑧 / 𝑥]𝜓
54 sbequ12 1729 . . . . . . . 8 (𝑥 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
5554dcbid 808 . . . . . . 7 (𝑥 = 𝑧 → (DECID 𝜓DECID [𝑧 / 𝑥]𝜓))
5653, 55rspc 2757 . . . . . 6 (𝑧𝐴 → (∀𝑥𝐴 DECID 𝜓DECID [𝑧 / 𝑥]𝜓))
5749, 51, 56sylc 62 . . . . 5 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → DECID [𝑧 / 𝑥]𝜓)
58 exmiddc 806 . . . . 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 772 . . 3 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
6160ex 114 . 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 6754 1 (𝜑 → {𝑥𝐴𝜓} ∈ Fin)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 682  DECID wdc 804   = wceq 1316  wcel 1465  [wsb 1720  wral 2393  {crab 2397  Vcvv 2660  [wsbc 2882  cdif 3038  cun 3039  wss 3041  c0 3333  {csn 3497  Fincfn 6602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1o 6281  df-er 6397  df-en 6603  df-fin 6605
This theorem is referenced by:  ssfidc  6791  phivalfi  11815  hashdvds  11824  phiprmpw  11825  phimullem  11828  hashgcdeq  11831
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