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Theorem ctiunct 12009
 Description: A sequence of enumerations gives an enumeration of the union. We refer to "sequence of enumerations" rather than "countably many countable sets" because the hypothesis provides more than countability for each : it refers to together with the which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74. The "countably many countable sets" version could be expressed as ⊔ and countable choice (or something similar) would be needed to derive the current hypothesis from that. Compare with the case of two sets instead of countably many, as seen at unct 12011, in which case we express countability using . The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 11964) and using the first number to map to an element of and the second number to map to an element of B(x) . In this way we are able to map to every element of . Although it would be possible to work directly with countability expressed as ⊔ , we instead use functions from subsets of the natural numbers via ctssdccl 7006 and ctssdc 7008. (Contributed by Jim Kingdon, 31-Oct-2023.)
Hypotheses
Ref Expression
ctiunct.a
ctiunct.b
Assertion
Ref Expression
ctiunct
Distinct variable groups:   ,,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   (,)

Proof of Theorem ctiunct
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpomen 11964 . . . . 5
21ensymi 6685 . . . 4
3 bren 6650 . . . 4
42, 3mpbi 144 . . 3
54a1i 9 . 2
6 ctiunct.a . . . . . . . 8
7 eqid 2140 . . . . . . . 8 inl inl
8 eqid 2140 . . . . . . . 8 inl inl
96, 7, 8ctssdccl 7006 . . . . . . 7 inl inl inl DECID inl
109simp1d 994 . . . . . 6 inl
1110adantr 274 . . . . 5 inl
129simp3d 996 . . . . . 6 DECID inl
1312adantr 274 . . . . 5 DECID inl
149simp2d 995 . . . . . 6 inl inl
1514adantr 274 . . . . 5 inl inl
16 ctiunct.b . . . . . . . 8
17 eqid 2140 . . . . . . . 8 inl inl
18 eqid 2140 . . . . . . . 8 inl inl
1916, 17, 18ctssdccl 7006 . . . . . . 7 inl inl inl DECID inl
2019simp1d 994 . . . . . 6 inl
2120adantlr 469 . . . . 5 inl
2219simp3d 996 . . . . . 6 DECID inl
2322adantlr 469 . . . . 5 DECID inl
2419simp2d 995 . . . . . 6 inl inl
2524adantlr 469 . . . . 5 inl inl
26 simpr 109 . . . . 5
27 eqid 2140 . . . . 5 inl inl inl inl inl inl
2811, 13, 15, 21, 23, 25, 26, 27ctiunctlemuom 12005 . . . 4 inl inl inl
29 eqid 2140 . . . . . 6 inl inl inl inl inl inl inl inl inl inl
30 nfv 1509 . . . . . . . . 9 inl
31 nfcsb1v 3041 . . . . . . . . . 10 inl inl
3231nfel2 2295 . . . . . . . . 9 inl inl
3330, 32nfan 1545 . . . . . . . 8 inl inl inl
34 nfcv 2282 . . . . . . . 8
3533, 34nfrabxy 2615 . . . . . . 7 inl inl inl
36 nfcsb1v 3041 . . . . . . . 8 inl inl
37 nfcv 2282 . . . . . . . 8
3836, 37nffv 5440 . . . . . . 7 inl inl
3935, 38nfmpt 4029 . . . . . 6 inl inl inl inl inl
4011, 13, 15, 21, 23, 25, 26, 27, 29, 39, 35ctiunctlemfo 12008 . . . . 5 inl inl inl inl inl inl inl inl
41 omex 4516 . . . . . . . 8
4241rabex 4081 . . . . . . 7 inl inl inl
4342mptex 5655 . . . . . 6 inl inl inl inl inl
44 foeq1 5350 . . . . . 6 inl inl inl inl inl inl inl inl inl inl inl inl inl inl inl inl
4543, 44spcev 2785 . . . . 5 inl inl inl inl inl inl inl inl inl inl inl
4640, 45syl 14 . . . 4 inl inl inl
4711, 13, 15, 21, 23, 25, 26, 27ctiunctlemudc 12006 . . . 4 DECID inl inl inl
48 sseq1 3126 . . . . . 6 inl inl inl inl inl inl
49 foeq2 5351 . . . . . . 7 inl inl inl inl inl inl
5049exbidv 1798 . . . . . 6 inl inl inl inl inl inl
51 eleq2 2204 . . . . . . . 8 inl inl inl inl inl inl
5251dcbid 824 . . . . . . 7 inl inl inl DECID DECID inl inl inl
5352ralbidv 2439 . . . . . 6 inl inl inl DECID DECID inl inl inl
5448, 50, 533anbi123d 1291 . . . . 5 inl inl inl DECID inl inl inl inl inl inl DECID inl inl inl
5542, 54spcev 2785 . . . 4 inl inl inl inl inl inl DECID inl inl inl DECID
5628, 46, 47, 55syl3anc 1217 . . 3 DECID
57 ctssdc 7008 . . . 4 DECID
58 foeq1 5350 . . . . 5
5958cbvexv 1891 . . . 4
6057, 59bitri 183 . . 3 DECID
6156, 60sylib 121 . 2
625, 61exlimddv 1871 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103  DECID wdc 820   w3a 963   wceq 1332  wex 1469   wcel 1481  wral 2417  crab 2421  csb 3008   wss 3077  ciun 3822   class class class wbr 3938   cmpt 3998  com 4513   cxp 4546  ccnv 4547  cima 4551   ccom 4552  wfo 5130  wf1o 5131  cfv 5132  c1st 6045  c2nd 6046  c1o 6315   cen 6641   ⊔ cdju 6932  inlcinl 6940 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-iinf 4511  ax-cnex 7755  ax-resscn 7756  ax-1cn 7757  ax-1re 7758  ax-icn 7759  ax-addcl 7760  ax-addrcl 7761  ax-mulcl 7762  ax-mulrcl 7763  ax-addcom 7764  ax-mulcom 7765  ax-addass 7766  ax-mulass 7767  ax-distr 7768  ax-i2m1 7769  ax-0lt1 7770  ax-1rid 7771  ax-0id 7772  ax-rnegex 7773  ax-precex 7774  ax-cnre 7775  ax-pre-ltirr 7776  ax-pre-ltwlin 7777  ax-pre-lttrn 7778  ax-pre-apti 7779  ax-pre-ltadd 7780  ax-pre-mulgt0 7781  ax-pre-mulext 7782  ax-arch 7783 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-xor 1355  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-if 3481  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-id 4224  df-po 4227  df-iso 4228  df-iord 4297  df-on 4299  df-ilim 4300  df-suc 4302  df-iom 4514  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-riota 5739  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-recs 6211  df-frec 6297  df-1o 6322  df-er 6438  df-en 6644  df-dju 6933  df-inl 6942  df-inr 6943  df-case 6979  df-pnf 7846  df-mnf 7847  df-xr 7848  df-ltxr 7849  df-le 7850  df-sub 7979  df-neg 7980  df-reap 8381  df-ap 8388  df-div 8477  df-inn 8765  df-2 8823  df-n0 9022  df-z 9099  df-uz 9371  df-q 9459  df-rp 9491  df-fz 9842  df-fl 10094  df-mod 10147  df-seqfrec 10270  df-exp 10344  df-dvds 11550 This theorem is referenced by:  ctiunctal  12010  unct  12011
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