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Theorem cantnflem4 8753
Description: Lemma for cantnf 8754. Complete the induction step of cantnflem3 8752. (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 8751 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))
9 eqid 2771 . . . . . . . . . . . . . 14 𝑋 = 𝑋
10 eqid 2771 . . . . . . . . . . . . . 14 𝑌 = 𝑌
11 eqid 2771 . . . . . . . . . . . . . 14 𝑍 = 𝑍
129, 10, 113pm3.2i 1423 . . . . . . . . . . . . 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 7836 . . . . . . . . . . . . 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 482 . . . . . . . . . 10 (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑍 ∈ (𝐴𝑜 𝑋)))
2120simp1d 1136 . . . . . . . . 9 (𝜑𝑋 ∈ On)
22 oecl 7771 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
232, 21, 22syl2anc 573 . . . . . . . 8 (𝜑 → (𝐴𝑜 𝑋) ∈ On)
2420simp2d 1137 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝐴 ∖ 1𝑜))
2524eldifad 3735 . . . . . . . . 9 (𝜑𝑌𝐴)
26 onelon 5891 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
272, 25, 26syl2anc 573 . . . . . . . 8 (𝜑𝑌 ∈ On)
28 omcl 7770 . . . . . . . 8 (((𝐴𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ∈ On)
2923, 27, 28syl2anc 573 . . . . . . 7 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ∈ On)
3020simp3d 1138 . . . . . . . 8 (𝜑𝑍 ∈ (𝐴𝑜 𝑋))
31 onelon 5891 . . . . . . . 8 (((𝐴𝑜 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴𝑜 𝑋)) → 𝑍 ∈ On)
3223, 30, 31syl2anc 573 . . . . . . 7 (𝜑𝑍 ∈ On)
33 oaword1 7786 . . . . . . 7 ((((𝐴𝑜 𝑋) ·𝑜 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ⊆ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍))
3429, 32, 33syl2anc 573 . . . . . 6 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 𝑌) ⊆ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍))
35 dif1o 7734 . . . . . . . . . . 11 (𝑌 ∈ (𝐴 ∖ 1𝑜) ↔ (𝑌𝐴𝑌 ≠ ∅))
3635simprbi 484 . . . . . . . . . 10 (𝑌 ∈ (𝐴 ∖ 1𝑜) → 𝑌 ≠ ∅)
3724, 36syl 17 . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
38 on0eln0 5923 . . . . . . . . . 10 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
3927, 38syl 17 . . . . . . . . 9 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
4037, 39mpbird 247 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑌)
41 omword1 7807 . . . . . . . 8 ((((𝐴𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴𝑜 𝑋) ⊆ ((𝐴𝑜 𝑋) ·𝑜 𝑌))
4223, 27, 40, 41syl21anc 1475 . . . . . . 7 (𝜑 → (𝐴𝑜 𝑋) ⊆ ((𝐴𝑜 𝑋) ·𝑜 𝑌))
4342, 30sseldd 3753 . . . . . 6 (𝜑𝑍 ∈ ((𝐴𝑜 𝑋) ·𝑜 𝑌))
4434, 43sseldd 3753 . . . . 5 (𝜑𝑍 ∈ (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍))
4519simprd 483 . . . . 5 (𝜑 → (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑍) = 𝐶)
4644, 45eleqtrd 2852 . . . 4 (𝜑𝑍𝐶)
471, 46sseldd 3753 . . 3 (𝜑𝑍 ∈ ran (𝐴 CNF 𝐵))
483, 2, 4cantnff 8735 . . . 4 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))
49 ffn 6185 . . . 4 ((𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
50 fvelrnb 6385 . . . 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 466 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐴 ∈ On)
544adantr 466 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐵 ∈ On)
556adantr 466 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ (𝐴𝑜 𝐵))
561adantr 466 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
577adantr 466 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ∅ ∈ 𝐶)
58 simprl 754 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝑔𝑆)
59 simprr 756 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
60 eqid 2771 . . 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 8752 . 2 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ ran (𝐴 CNF 𝐵))
6252, 61rexlimddv 3183 1 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  cdif 3720  wss 3723  c0 4063  ifcif 4225  cop 4322   cuni 4574   cint 4611  {copab 4846  cmpt 4863  dom cdm 5249  ran crn 5250  Oncon0 5866  cio 5992   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  1𝑜c1o 7706  2𝑜c2o 7707   +𝑜 coa 7710   ·𝑜 comu 7711  𝑜 coe 7712   CNF ccnf 8722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-oexp 7719  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-oi 8571  df-cnf 8723
This theorem is referenced by:  cantnf  8754
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