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Theorem ctssdclemr 7110
Description: Lemma for ctssdc 7111. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)
Assertion
Ref Expression
ctssdclemr (āˆƒš‘“ š‘“:Ļ‰ā€“ontoā†’(š“ āŠ” 1o) ā†’ āˆƒš‘ (š‘  āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:š‘ ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ š‘ ))
Distinct variable groups:   š“,š‘“,š‘    š“,š‘›,š‘ 

Proof of Theorem ctssdclemr
Dummy variables š‘” š‘” are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 foeq1 5434 . . 3 (š‘“ = š‘” ā†’ (š‘“:Ļ‰ā€“ontoā†’(š“ āŠ” 1o) ā†” š‘”:Ļ‰ā€“ontoā†’(š“ āŠ” 1o)))
21cbvexv 1918 . 2 (āˆƒš‘“ š‘“:Ļ‰ā€“ontoā†’(š“ āŠ” 1o) ā†” āˆƒš‘” š‘”:Ļ‰ā€“ontoā†’(š“ āŠ” 1o))
3 id 19 . . . . . 6 (š‘”:Ļ‰ā€“ontoā†’(š“ āŠ” 1o) ā†’ š‘”:Ļ‰ā€“ontoā†’(š“ āŠ” 1o))
4 eqid 2177 . . . . . 6 {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} = {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}
5 eqid 2177 . . . . . 6 (ā—”inl āˆ˜ š‘”) = (ā—”inl āˆ˜ š‘”)
63, 4, 5ctssdccl 7109 . . . . 5 (š‘”:Ļ‰ā€“ontoā†’(š“ āŠ” 1o) ā†’ ({š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} āŠ† Ļ‰ āˆ§ (ā—”inl āˆ˜ š‘”):{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}))
7 djulf1o 7056 . . . . . . . . 9 inl:Vā€“1-1-ontoā†’({āˆ…} Ɨ V)
8 f1ocnv 5474 . . . . . . . . 9 (inl:Vā€“1-1-ontoā†’({āˆ…} Ɨ V) ā†’ ā—”inl:({āˆ…} Ɨ V)ā€“1-1-ontoā†’V)
9 f1ofun 5463 . . . . . . . . 9 (ā—”inl:({āˆ…} Ɨ V)ā€“1-1-ontoā†’V ā†’ Fun ā—”inl)
107, 8, 9mp2b 8 . . . . . . . 8 Fun ā—”inl
11 vex 2740 . . . . . . . 8 š‘” āˆˆ V
12 cofunexg 6109 . . . . . . . 8 ((Fun ā—”inl āˆ§ š‘” āˆˆ V) ā†’ (ā—”inl āˆ˜ š‘”) āˆˆ V)
1310, 11, 12mp2an 426 . . . . . . 7 (ā—”inl āˆ˜ š‘”) āˆˆ V
14 foeq1 5434 . . . . . . 7 (š‘“ = (ā—”inl āˆ˜ š‘”) ā†’ (š‘“:{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“ ā†” (ā—”inl āˆ˜ š‘”):{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“))
1513, 14spcev 2832 . . . . . 6 ((ā—”inl āˆ˜ š‘”):{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“ ā†’ āˆƒš‘“ š‘“:{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“)
16153anim2i 1186 . . . . 5 (({š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} āŠ† Ļ‰ āˆ§ (ā—”inl āˆ˜ š‘”):{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}) ā†’ ({š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}))
176, 16syl 14 . . . 4 (š‘”:Ļ‰ā€“ontoā†’(š“ āŠ” 1o) ā†’ ({š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}))
18 omex 4592 . . . . . 6 Ļ‰ āˆˆ V
1918rabex 4147 . . . . 5 {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} āˆˆ V
20 sseq1 3178 . . . . . 6 (š‘  = {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} ā†’ (š‘  āŠ† Ļ‰ ā†” {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} āŠ† Ļ‰))
21 foeq2 5435 . . . . . . 7 (š‘  = {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} ā†’ (š‘“:š‘ ā€“ontoā†’š“ ā†” š‘“:{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“))
2221exbidv 1825 . . . . . 6 (š‘  = {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} ā†’ (āˆƒš‘“ š‘“:š‘ ā€“ontoā†’š“ ā†” āˆƒš‘“ š‘“:{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“))
23 eleq2 2241 . . . . . . . 8 (š‘  = {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} ā†’ (š‘› āˆˆ š‘  ā†” š‘› āˆˆ {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}))
2423dcbid 838 . . . . . . 7 (š‘  = {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} ā†’ (DECID š‘› āˆˆ š‘  ā†” DECID š‘› āˆˆ {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}))
2524ralbidv 2477 . . . . . 6 (š‘  = {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} ā†’ (āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ š‘  ā†” āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}))
2620, 22, 253anbi123d 1312 . . . . 5 (š‘  = {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} ā†’ ((š‘  āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:š‘ ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ š‘ ) ā†” ({š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)})))
2719, 26spcev 2832 . . . 4 (({š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)} āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:{š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ {š‘” āˆˆ Ļ‰ āˆ£ (š‘”ā€˜š‘”) āˆˆ (inl ā€œ š“)}) ā†’ āˆƒš‘ (š‘  āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:š‘ ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ š‘ ))
2817, 27syl 14 . . 3 (š‘”:Ļ‰ā€“ontoā†’(š“ āŠ” 1o) ā†’ āˆƒš‘ (š‘  āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:š‘ ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ š‘ ))
2928exlimiv 1598 . 2 (āˆƒš‘” š‘”:Ļ‰ā€“ontoā†’(š“ āŠ” 1o) ā†’ āˆƒš‘ (š‘  āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:š‘ ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ š‘ ))
302, 29sylbi 121 1 (āˆƒš‘“ š‘“:Ļ‰ā€“ontoā†’(š“ āŠ” 1o) ā†’ āˆƒš‘ (š‘  āŠ† Ļ‰ āˆ§ āˆƒš‘“ š‘“:š‘ ā€“ontoā†’š“ āˆ§ āˆ€š‘› āˆˆ Ļ‰ DECID š‘› āˆˆ š‘ ))
Colors of variables: wff set class
Syntax hints:   ā†’ wi 4  DECID wdc 834   āˆ§ w3a 978   = wceq 1353  āˆƒwex 1492   āˆˆ wcel 2148  āˆ€wral 2455  {crab 2459  Vcvv 2737   āŠ† wss 3129  āˆ…c0 3422  {csn 3592  Ļ‰com 4589   Ɨ cxp 4624  ā—”ccnv 4625   ā€œ cima 4629   āˆ˜ ccom 4630  Fun wfun 5210  ā€“ontoā†’wfo 5214  ā€“1-1-ontoā†’wf1o 5215  ā€˜cfv 5216  1oc1o 6409   āŠ” cdju 7035  inlcinl 7043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-dju 7036  df-inl 7045  df-inr 7046
This theorem is referenced by:  ctssdc  7111
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