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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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