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Theorem cantnflem4 8536
Description: Lemma for cantnf 8537. Complete the induction step of cantnflem3 8535. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
cantnf.c (𝜑𝐶 ∈ (𝐴𝑜 𝐵))
cantnf.s (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))
cantnf.e (𝜑 → ∅ ∈ 𝐶)
cantnf.x 𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴𝑜 𝑐)}
cantnf.p 𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑎) +𝑜 𝑏) = 𝐶))
cantnf.y 𝑌 = (1st𝑃)
cantnf.z 𝑍 = (2nd𝑃)
Assertion
Ref Expression
cantnflem4 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Distinct variable groups:   𝑤,𝑐,𝑥,𝑦,𝑧,𝐵   𝑎,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧,𝐶   𝐴,𝑎,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐   𝑆,𝑐,𝑥,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑤,𝑌,𝑥,𝑦,𝑧   𝑋,𝑎,𝑏,𝑑,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑤,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑎,𝑏,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏,𝑐,𝑑)   𝑆(𝑤,𝑎,𝑏,𝑑)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏,𝑑)   𝑋(𝑐)   𝑌(𝑎,𝑏,𝑐,𝑑)   𝑍(𝑤,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem cantnflem4
Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnf.s . . . 4 (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))
2 cantnfs.a . . . . . . . . 9 (𝜑𝐴 ∈ On)
3 cantnfs.s . . . . . . . . . . . . 13 𝑆 = dom (𝐴 CNF 𝐵)
4 cantnfs.b . . . . . . . . . . . . 13 (𝜑𝐵 ∈ On)
5 oemapval.t . . . . . . . . . . . . 13 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
6 cantnf.c . . . . . . . . . . . . 13 (𝜑𝐶 ∈ (𝐴𝑜 𝐵))
7 cantnf.e . . . . . . . . . . . . 13 (𝜑 → ∅ ∈ 𝐶)
83, 2, 4, 5, 6, 1, 7cantnflem2 8534 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))
9 eqid 2621 . . . . . . . . . . . . . 14 𝑋 = 𝑋
10 eqid 2621 . . . . . . . . . . . . . 14 𝑌 = 𝑌
11 eqid 2621 . . . . . . . . . . . . . 14 𝑍 = 𝑍
129, 10, 113pm3.2i 1237 . . . . . . . . . . . . 13 (𝑋 = 𝑋𝑌 = 𝑌𝑍 = 𝑍)
13 cantnf.x . . . . . . . . . . . . . 14 𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴𝑜 𝑐)}
14 cantnf.p . . . . . . . . . . . . . 14 𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴𝑜 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑎) +𝑜 𝑏) = 𝐶))
15 cantnf.y . . . . . . . . . . . . . 14 𝑌 = (1st𝑃)
16 cantnf.z . . . . . . . . . . . . . 14 𝑍 = (2nd𝑃)
1713, 14, 15, 16oeeui 7630 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)) → (((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑍 ∈ (𝐴𝑜 𝑋)) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍) = 𝐶) ↔ (𝑋 = 𝑋𝑌 = 𝑌𝑍 = 𝑍)))
1812, 17mpbiri 248 . . . . . . . . . . . 12 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)) → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑍 ∈ (𝐴𝑜 𝑋)) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍) = 𝐶))
198, 18syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑍 ∈ (𝐴𝑜 𝑋)) ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍) = 𝐶))
2019simpld 475 . . . . . . . . . 10 (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑍 ∈ (𝐴𝑜 𝑋)))
2120simp1d 1071 . . . . . . . . 9 (𝜑𝑋 ∈ On)
22 oecl 7565 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
232, 21, 22syl2anc 692 . . . . . . . 8 (𝜑 → (𝐴𝑜 𝑋) ∈ On)
2420simp2d 1072 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝐴 ∖ 1𝑜))
2524eldifad 3568 . . . . . . . . 9 (𝜑𝑌𝐴)
26 onelon 5709 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
272, 25, 26syl2anc 692 . . . . . . . 8 (𝜑𝑌 ∈ On)
28 omcl 7564 . . . . . . . 8 (((𝐴𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ∈ On)
2923, 27, 28syl2anc 692 . . . . . . 7 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ∈ On)
3020simp3d 1073 . . . . . . . 8 (𝜑𝑍 ∈ (𝐴𝑜 𝑋))
31 onelon 5709 . . . . . . . 8 (((𝐴𝑜 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴𝑜 𝑋)) → 𝑍 ∈ On)
3223, 30, 31syl2anc 692 . . . . . . 7 (𝜑𝑍 ∈ On)
33 oaword1 7580 . . . . . . 7 ((((𝐴𝑜 𝑋) ·𝑜 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ⊆ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍))
3429, 32, 33syl2anc 692 . . . . . 6 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ⊆ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍))
35 dif1o 7528 . . . . . . . . . . 11 (𝑌 ∈ (𝐴 ∖ 1𝑜) ↔ (𝑌𝐴𝑌 ≠ ∅))
3635simprbi 480 . . . . . . . . . 10 (𝑌 ∈ (𝐴 ∖ 1𝑜) → 𝑌 ≠ ∅)
3724, 36syl 17 . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
38 on0eln0 5741 . . . . . . . . . 10 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
3927, 38syl 17 . . . . . . . . 9 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
4037, 39mpbird 247 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑌)
41 omword1 7601 . . . . . . . 8 ((((𝐴𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴𝑜 𝑋) ⊆ ((𝐴𝑜 𝑋) ·𝑜 𝑌))
4223, 27, 40, 41syl21anc 1322 . . . . . . 7 (𝜑 → (𝐴𝑜 𝑋) ⊆ ((𝐴𝑜 𝑋) ·𝑜 𝑌))
4342, 30sseldd 3585 . . . . . 6 (𝜑𝑍 ∈ ((𝐴𝑜 𝑋) ·𝑜 𝑌))
4434, 43sseldd 3585 . . . . 5 (𝜑𝑍 ∈ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍))
4519simprd 479 . . . . 5 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍) = 𝐶)
4644, 45eleqtrd 2700 . . . 4 (𝜑𝑍𝐶)
471, 46sseldd 3585 . . 3 (𝜑𝑍 ∈ ran (𝐴 CNF 𝐵))
483, 2, 4cantnff 8518 . . . 4 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))
49 ffn 6004 . . . 4 ((𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
50 fvelrnb 6202 . . . 4 ((𝐴 CNF 𝐵) Fn 𝑆 → (𝑍 ∈ ran (𝐴 CNF 𝐵) ↔ ∃𝑔𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍))
5148, 49, 503syl 18 . . 3 (𝜑 → (𝑍 ∈ ran (𝐴 CNF 𝐵) ↔ ∃𝑔𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍))
5247, 51mpbid 222 . 2 (𝜑 → ∃𝑔𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
532adantr 481 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐴 ∈ On)
544adantr 481 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐵 ∈ On)
556adantr 481 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ (𝐴𝑜 𝐵))
561adantr 481 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
577adantr 481 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ∅ ∈ 𝐶)
58 simprl 793 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝑔𝑆)
59 simprr 795 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
60 eqid 2621 . . 3 (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝑔𝑡))) = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝑔𝑡)))
613, 53, 54, 5, 55, 56, 57, 13, 14, 15, 16, 58, 59, 60cantnflem3 8535 . 2 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ ran (𝐴 CNF 𝐵))
6252, 61rexlimddv 3028 1 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cdif 3553  wss 3556  c0 3893  ifcif 4060  cop 4156   cuni 4404   cint 4442  {copab 4674  cmpt 4675  dom cdm 5076  ran crn 5077  Oncon0 5684  cio 5810   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607  1st c1st 7114  2nd c2nd 7115  1𝑜c1o 7501  2𝑜c2o 7502   +𝑜 coa 7505   ·𝑜 comu 7506  𝑜 coe 7507   CNF ccnf 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-supp 7244  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-seqom 7491  df-1o 7508  df-2o 7509  df-oadd 7512  df-omul 7513  df-oexp 7514  df-er 7690  df-map 7807  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-fsupp 8223  df-oi 8362  df-cnf 8506
This theorem is referenced by:  cantnf  8537
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