Theorem cantnflem4 9645

Description: Lemma for cantnf 9646. Complete the induction step of cantnflem3 9644. (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 9643	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
9		eqid 2729	. . . . . . . . . . . . . 14 ⊢ 𝑋 = 𝑋
10		eqid 2729	. . . . . . . . . . . . . 14 ⊢ 𝑌 = 𝑌
11		eqid 2729	. . . . . . . . . . . . . 14 ⊢ 𝑍 = 𝑍
12	9, 10, 11	3pm3.2i 1340	. . . . . . . . . . . . 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 8566	. . . . . . . . . . . . 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 258	. . . . . . . . . . . 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 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ On)
22		oecl 8501	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
23	2, 21, 22	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ∈ On)
24	20	simp2d 1143	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 1o))
25	24	eldifad 3926	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
26		onelon 6357	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ On)
27	2, 25, 26	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ On)
28		omcl 8500	. . . . . . . 8 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On)
29	23, 27, 28	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On)
30	20	simp3d 1144	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝐴 ↑o 𝑋))
31		onelon 6357	. . . . . . . 8 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) → 𝑍 ∈ On)
32	23, 30, 31	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ On)
33		oaword1 8516	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o 𝑌) ⊆ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
35		dif1o 8464	. . . . . . . . . . 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 6389	. . . . . . . . . 10 ⊢ (𝑌 ∈ On → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
39	27, 38	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
40	37, 39	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝑌)
41		omword1 8537	. . . . . . . 8 ⊢ ((((𝐴 ↑o 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴 ↑o 𝑋) ⊆ ((𝐴 ↑o 𝑋) ·o 𝑌))
42	23, 27, 40, 41	syl21anc 837	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ⊆ ((𝐴 ↑o 𝑋) ·o 𝑌))
43	42, 30	sseldd 3947	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ((𝐴 ↑o 𝑋) ·o 𝑌))
44	34, 43	sseldd 3947	. . . . 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 2830	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
47	1, 46	sseldd 3947	. . 3 ⊢ (𝜑 → 𝑍 ∈ ran (𝐴 CNF 𝐵))
48	3, 2, 4	cantnff 9627	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵))
49		ffn 6688	. . . 4 ⊢ ((𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
50		fvelrnb 6921	. . . 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 232	. 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 770	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝑔 ∈ 𝑆)
59		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
60		eqid 2729	. . 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 9644	. 2 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ ran (𝐴 CNF 𝐵))
62	52, 61	rexlimddv 3140	1 ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3405   ∖ cdif 3911   ⊆ wss 3914  ∅c0 4296  ifcif 4488  ⟨cop 4595  ∪ cuni 4871  ∩ cint 4910  {copab 5169   ↦ cmpt 5188  dom cdm 5638  ran crn 5639  Oncon0 6332  ℩cio 6462   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  1_oc1o 8427  2_oc2o 8428   +_o coa 8431   ·_o comu 8432   ↑_o coe 8433   CNF ccnf 9614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-seqom 8416  df-1o 8434  df-2o 8435  df-oadd 8438  df-omul 8439  df-oexp 8440  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-oi 9463  df-cnf 9615

This theorem is referenced by:  cantnf  9646