Theorem cantnflem4 9380

Description: Lemma for cantnf 9381. Complete the induction step of cantnflem3 9379. (Contributed by Mario Carneiro, 25-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
cantnf.c	⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))
cantnf.s	⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
cantnf.e	⊢ (𝜑 → ∅ ∈ 𝐶)
cantnf.x	⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}
cantnf.p	⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
cantnf.y	⊢ 𝑌 = (1st ‘𝑃)
cantnf.z	⊢ 𝑍 = (2nd ‘𝑃)

Assertion

Ref	Expression
cantnflem4	⊢ (𝜑 → 𝐶 ∈ 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.

Step	Hyp	Ref	Expression
1		cantnf.s	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
2		cantnfs.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ On)
3		cantnfs.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
4		cantnfs.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ On)
5		oemapval.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
6		cantnf.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))
7		cantnf.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ 𝐶)
8	3, 2, 4, 5, 6, 1, 7	cantnflem2 9378	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
9		eqid 2738	. . . . . . . . . . . . . 14 ⊢ 𝑋 = 𝑋
10		eqid 2738	. . . . . . . . . . . . . 14 ⊢ 𝑌 = 𝑌
11		eqid 2738	. . . . . . . . . . . . . 14 ⊢ 𝑍 = 𝑍
12	9, 10, 11	3pm3.2i 1337	. . . . . . . . . . . . 13 ⊢ (𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍)
13		cantnf.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}
14		cantnf.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
15		cantnf.y	. . . . . . . . . . . . . 14 ⊢ 𝑌 = (1st ‘𝑃)
16		cantnf.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (2nd ‘𝑃)
17	13, 14, 15, 16	oeeui 8395	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → (((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) ∧ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶) ↔ (𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍)))
18	12, 17	mpbiri 257	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) ∧ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
19	8, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) ∧ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
20	19	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)))
21	20	simp1d 1140	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ On)
22		oecl 8329	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
23	2, 21, 22	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ∈ On)
24	20	simp2d 1141	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 1o))
25	24	eldifad 3895	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
26		onelon 6276	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ On)
27	2, 25, 26	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ On)
28		omcl 8328	. . . . . . . 8 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On)
29	23, 27, 28	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On)
30	20	simp3d 1142	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝐴 ↑o 𝑋))
31		onelon 6276	. . . . . . . 8 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) → 𝑍 ∈ On)
32	23, 30, 31	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ On)
33		oaword1 8345	. . . . . . 7 ⊢ ((((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴 ↑o 𝑋) ·o 𝑌) ⊆ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
34	29, 32, 33	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o 𝑌) ⊆ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
35		dif1o 8292	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (𝐴 ∖ 1o) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ≠ ∅))
36	35	simprbi 496	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐴 ∖ 1o) → 𝑌 ≠ ∅)
37	24, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ ∅)
38		on0eln0 6306	. . . . . . . . . 10 ⊢ (𝑌 ∈ On → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
39	27, 38	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
40	37, 39	mpbird 256	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝑌)
41		omword1 8366	. . . . . . . 8 ⊢ ((((𝐴 ↑o 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴 ↑o 𝑋) ⊆ ((𝐴 ↑o 𝑋) ·o 𝑌))
42	23, 27, 40, 41	syl21anc 834	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ⊆ ((𝐴 ↑o 𝑋) ·o 𝑌))
43	42, 30	sseldd 3918	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ((𝐴 ↑o 𝑋) ·o 𝑌))
44	34, 43	sseldd 3918	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
45	19	simprd 495	. . . . 5 ⊢ (𝜑 → (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶)
46	44, 45	eleqtrd 2841	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
47	1, 46	sseldd 3918	. . 3 ⊢ (𝜑 → 𝑍 ∈ ran (𝐴 CNF 𝐵))
48	3, 2, 4	cantnff 9362	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵))
49		ffn 6584	. . . 4 ⊢ ((𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
50		fvelrnb 6812	. . . 4 ⊢ ((𝐴 CNF 𝐵) Fn 𝑆 → (𝑍 ∈ ran (𝐴 CNF 𝐵) ↔ ∃𝑔 ∈ 𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍))
51	48, 49, 50	3syl 18	. . 3 ⊢ (𝜑 → (𝑍 ∈ ran (𝐴 CNF 𝐵) ↔ ∃𝑔 ∈ 𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍))
52	47, 51	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
53	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐴 ∈ On)
54	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐵 ∈ On)
55	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ (𝐴 ↑o 𝐵))
56	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
57	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ∅ ∈ 𝐶)
58		simprl 767	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝑔 ∈ 𝑆)
59		simprr 769	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
60		eqid 2738	. . 3 ⊢ (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝑔‘𝑡))) = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝑔‘𝑡)))
61	3, 53, 54, 5, 55, 56, 57, 13, 14, 15, 16, 58, 59, 60	cantnflem3 9379	. 2 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ ran (𝐴 CNF 𝐵))
62	52, 61	rexlimddv 3219	1 ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  ifcif 4456  ⟨cop 4564  ∪ cuni 4836  ∩ cint 4876  {copab 5132   ↦ cmpt 5153  dom cdm 5580  ran crn 5581  Oncon0 6251  ℩cio 6374   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  1_oc1o 8260  2_oc2o 8261   +_o coa 8264   ·_o comu 8265   ↑_o coe 8266   CNF ccnf 9349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-seqom 8249  df-1o 8267  df-2o 8268  df-oadd 8271  df-omul 8272  df-oexp 8273  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-oi 9199  df-cnf 9350

This theorem is referenced by:  cantnf  9381