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Theorem cantnflem4 9628
Description: Lemma for cantnf 9629. Complete the induction step of cantnflem3 9627. (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 9626 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
9 eqid 2736 . . . . . . . . . . . . . 14 𝑋 = 𝑋
10 eqid 2736 . . . . . . . . . . . . . 14 𝑌 = 𝑌
11 eqid 2736 . . . . . . . . . . . . . 14 𝑍 = 𝑍
129, 10, 113pm3.2i 1339 . . . . . . . . . . . . 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 8549 . . . . . . . . . . . . 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 257 . . . . . . . . . . . 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 495 . . . . . . . . . 10 (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)))
2120simp1d 1142 . . . . . . . . 9 (𝜑𝑋 ∈ On)
22 oecl 8483 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
232, 21, 22syl2anc 584 . . . . . . . 8 (𝜑 → (𝐴o 𝑋) ∈ On)
2420simp2d 1143 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝐴 ∖ 1o))
2524eldifad 3922 . . . . . . . . 9 (𝜑𝑌𝐴)
26 onelon 6342 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
272, 25, 26syl2anc 584 . . . . . . . 8 (𝜑𝑌 ∈ On)
28 omcl 8482 . . . . . . . 8 (((𝐴o 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
2923, 27, 28syl2anc 584 . . . . . . 7 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
3020simp3d 1144 . . . . . . . 8 (𝜑𝑍 ∈ (𝐴o 𝑋))
31 onelon 6342 . . . . . . . 8 (((𝐴o 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴o 𝑋)) → 𝑍 ∈ On)
3223, 30, 31syl2anc 584 . . . . . . 7 (𝜑𝑍 ∈ On)
33 oaword1 8499 . . . . . . 7 ((((𝐴o 𝑋) ·o 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴o 𝑋) ·o 𝑌) ⊆ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
3429, 32, 33syl2anc 584 . . . . . 6 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ⊆ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
35 dif1o 8446 . . . . . . . . . . 11 (𝑌 ∈ (𝐴 ∖ 1o) ↔ (𝑌𝐴𝑌 ≠ ∅))
3635simprbi 497 . . . . . . . . . 10 (𝑌 ∈ (𝐴 ∖ 1o) → 𝑌 ≠ ∅)
3724, 36syl 17 . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
38 on0eln0 6373 . . . . . . . . . 10 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
3927, 38syl 17 . . . . . . . . 9 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
4037, 39mpbird 256 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑌)
41 omword1 8520 . . . . . . . 8 ((((𝐴o 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴o 𝑋) ⊆ ((𝐴o 𝑋) ·o 𝑌))
4223, 27, 40, 41syl21anc 836 . . . . . . 7 (𝜑 → (𝐴o 𝑋) ⊆ ((𝐴o 𝑋) ·o 𝑌))
4342, 30sseldd 3945 . . . . . 6 (𝜑𝑍 ∈ ((𝐴o 𝑋) ·o 𝑌))
4434, 43sseldd 3945 . . . . 5 (𝜑𝑍 ∈ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
4519simprd 496 . . . . 5 (𝜑 → (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶)
4644, 45eleqtrd 2840 . . . 4 (𝜑𝑍𝐶)
471, 46sseldd 3945 . . 3 (𝜑𝑍 ∈ ran (𝐴 CNF 𝐵))
483, 2, 4cantnff 9610 . . . 4 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
49 ffn 6668 . . . 4 ((𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
50 fvelrnb 6903 . . . 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 231 . 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 769 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝑔𝑆)
59 simprr 771 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
60 eqid 2736 . . 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 9627 . 2 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ ran (𝐴 CNF 𝐵))
6252, 61rexlimddv 3158 1 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  cdif 3907  wss 3910  c0 4282  ifcif 4486  cop 4592   cuni 4865   cint 4907  {copab 5167  cmpt 5188  dom cdm 5633  ran crn 5634  Oncon0 6317  cio 6446   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  1oc1o 8405  2oc2o 8406   +o coa 8409   ·o comu 8410  o coe 8411   CNF ccnf 9597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-seqom 8394  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-oexp 8418  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-cnf 9598
This theorem is referenced by:  cantnf  9629
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