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Theorem ssfirab 6822
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 2678 . . 3 (𝑤 = ∅ → {𝑥𝑤𝜓} = {𝑥 ∈ ∅ ∣ 𝜓})
21eleq1d 2208 . 2 (𝑤 = ∅ → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin))
3 rabeq 2678 . . 3 (𝑤 = 𝑦 → {𝑥𝑤𝜓} = {𝑥𝑦𝜓})
43eleq1d 2208 . 2 (𝑤 = 𝑦 → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥𝑦𝜓} ∈ Fin))
5 rabeq 2678 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥𝑤𝜓} = {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓})
65eleq1d 2208 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin))
7 rabeq 2678 . . 3 (𝑤 = 𝐴 → {𝑥𝑤𝜓} = {𝑥𝐴𝜓})
87eleq1d 2208 . 2 (𝑤 = 𝐴 → ({𝑥𝑤𝜓} ∈ Fin ↔ {𝑥𝐴𝜓} ∈ Fin))
9 rab0 3391 . . . 4 {𝑥 ∈ ∅ ∣ 𝜓} = ∅
10 0fin 6778 . . . 4 ∅ ∈ Fin
119, 10eqeltri 2212 . . 3 {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin
1211a1i 9 . 2 (𝜑 → {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin)
13 rabun2 3355 . . . . 5 {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓})
14 sbsbc 2913 . . . . . . . . . 10 ([𝑧 / 𝑥]𝜓[𝑧 / 𝑥]𝜓)
15 vex 2689 . . . . . . . . . . 11 𝑧 ∈ V
16 ralsns 3562 . . . . . . . . . . 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 2607 . . . . . . . . 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 3230 . . . . . 6 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑧}) = ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}))
23 simplr 519 . . . . . . 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 521 . . . . . . . . . 10 (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
2625ad2antrr 479 . . . . . . . . 9 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → 𝑧 ∈ (𝐴𝑦))
2726eldifbd 3083 . . . . . . . 8 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧𝑦)
28 elrabi 2837 . . . . . . . 8 (𝑧 ∈ {𝑥𝑦𝜓} → 𝑧𝑦)
2927, 28nsyl 617 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧 ∈ {𝑥𝑦𝜓})
30 unsnfi 6807 . . . . . . 7 (({𝑥𝑦𝜓} ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ {𝑥𝑦𝜓}) → ({𝑥𝑦𝜓} ∪ {𝑧}) ∈ Fin)
3123, 24, 29, 30syl3anc 1216 . . . . . 6 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑧}) ∈ Fin)
3222, 31eqeltrrd 2217 . . . . 5 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) ∈ Fin)
3313, 32eqeltrid 2226 . . . 4 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
34 ralsns 3562 . . . . . . . . . . . 12 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧} ¬ 𝜓[𝑧 / 𝑥] ¬ 𝜓))
3515, 34ax-mp 5 . . . . . . . . . . 11 (∀𝑥 ∈ {𝑧} ¬ 𝜓[𝑧 / 𝑥] ¬ 𝜓)
36 sbsbc 2913 . . . . . . . . . . 11 ([𝑧 / 𝑥] ¬ 𝜓[𝑧 / 𝑥] ¬ 𝜓)
37 sbn 1925 . . . . . . . . . . 11 ([𝑧 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑧 / 𝑥]𝜓)
3835, 36, 373bitr2ri 208 . . . . . . . . . 10 (¬ [𝑧 / 𝑥]𝜓 ↔ ∀𝑥 ∈ {𝑧} ¬ 𝜓)
39 rabeq0 3392 . . . . . . . . . 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 3230 . . . . . . 7 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) = ({𝑥𝑦𝜓} ∪ ∅))
43 un0 3396 . . . . . . 7 ({𝑥𝑦𝜓} ∪ ∅) = {𝑥𝑦𝜓}
4442, 43syl6eq 2188 . . . . . 6 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → ({𝑥𝑦𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) = {𝑥𝑦𝜓})
4513, 44syl5eq 2184 . . . . 5 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = {𝑥𝑦𝜓})
46 simplr 519 . . . . 5 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥𝑦𝜓} ∈ Fin)
4745, 46eqeltrd 2216 . . . 4 (((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
48 simplrr 525 . . . . . . 7 ((((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ {𝑥𝑦𝜓} ∈ Fin) → 𝑧 ∈ (𝐴𝑦))
4948eldifad 3082 . . . . . 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 1912 . . . . . . . 8 𝑥[𝑧 / 𝑥]𝜓
5352nfdc 1637 . . . . . . 7 𝑥DECID [𝑧 / 𝑥]𝜓
54 sbequ12 1744 . . . . . . . 8 (𝑥 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
5554dcbid 823 . . . . . . 7 (𝑥 = 𝑧 → (DECID 𝜓DECID [𝑧 / 𝑥]𝜓))
5653, 55rspc 2783 . . . . . 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 821 . . . . 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 787 . . 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 6786 1 (𝜑 → {𝑥𝐴𝜓} ∈ Fin)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  DECID wdc 819   = wceq 1331  wcel 1480  [wsb 1735  wral 2416  {crab 2420  Vcvv 2686  [wsbc 2909  cdif 3068  cun 3069  wss 3071  c0 3363  {csn 3527  Fincfn 6634
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1o 6313  df-er 6429  df-en 6635  df-fin 6637
This theorem is referenced by:  ssfidc  6823  phivalfi  11899  hashdvds  11908  phiprmpw  11909  phimullem  11912  hashgcdeq  11915
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