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Theorem cantnflem4 8589

Description: Lemma for cantnf 8590. Complete the induction step of cantnflem3 8588. (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 = (1^st ‘𝑃)
cantnf.z	⊢ 𝑍 = (2^nd ‘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 8587	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐶 ∈ (On ∖ 1_𝑜)))
9		eqid 2622	. . . . . . . . . . . . . 14 ⊢ 𝑋 = 𝑋
10		eqid 2622	. . . . . . . . . . . . . 14 ⊢ 𝑌 = 𝑌
11		eqid 2622	. . . . . . . . . . . . . 14 ⊢ 𝑍 = 𝑍
12	9, 10, 11	3pm3.2i 1239	. . . . . . . . . . . . 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 = (1^st ‘𝑃)
16		cantnf.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (2^nd ‘𝑃)
17	13, 14, 15, 16	oeeui 7682	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐶 ∈ (On ∖ 1_𝑜)) → (((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) ∧ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍) = 𝐶) ↔ (𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍)))
18	12, 17	mpbiri 248	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐶 ∈ (On ∖ 1_𝑜)) → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) ∧ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍) = 𝐶))
19	8, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) ∧ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍) = 𝐶))
20	19	simpld 475	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)))
21	20	simp1d 1073	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ On)
22		oecl 7617	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑_𝑜 𝑋) ∈ On)
23	2, 21, 22	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ∈ On)
24	20	simp2d 1074	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 1_𝑜))
25	24	eldifad 3586	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
26		onelon 5748	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ On)
27	2, 25, 26	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ On)
28		omcl 7616	. . . . . . . 8 ⊢ (((𝐴 ↑_𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ∈ On)
29	23, 27, 28	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ∈ On)
30	20	simp3d 1075	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝐴 ↑_𝑜 𝑋))
31		onelon 5748	. . . . . . . 8 ⊢ (((𝐴 ↑_𝑜 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) → 𝑍 ∈ On)
32	23, 30, 31	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ On)
33		oaword1 7632	. . . . . . 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 693	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ⊆ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍))
35		dif1o 7580	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (𝐴 ∖ 1_𝑜) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ≠ ∅))
36	35	simprbi 480	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐴 ∖ 1_𝑜) → 𝑌 ≠ ∅)
37	24, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ ∅)
38		on0eln0 5780	. . . . . . . . . 10 ⊢ (𝑌 ∈ On → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
39	27, 38	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
40	37, 39	mpbird 247	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝑌)
41		omword1 7653	. . . . . . . 8 ⊢ ((((𝐴 ↑_𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴 ↑_𝑜 𝑋) ⊆ ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌))
42	23, 27, 40, 41	syl21anc 1325	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ⊆ ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌))
43	42, 30	sseldd 3604	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌))
44	34, 43	sseldd 3604	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍))
45	19	simprd 479	. . . . 5 ⊢ (𝜑 → (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍) = 𝐶)
46	44, 45	eleqtrd 2703	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
47	1, 46	sseldd 3604	. . 3 ⊢ (𝜑 → 𝑍 ∈ ran (𝐴 CNF 𝐵))
48	3, 2, 4	cantnff 8571	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵))
49		ffn 6045	. . . 4 ⊢ ((𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
50		fvelrnb 6243	. . . 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 222	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
53	2	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐴 ∈ On)
54	4	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐵 ∈ On)
55	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ (𝐴 ↑_𝑜 𝐵))
56	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
57	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ∅ ∈ 𝐶)
58		simprl 794	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝑔 ∈ 𝑆)
59		simprr 796	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
60		eqid 2622	. . 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 8588	. 2 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ ran (𝐴 CNF 𝐵))
62	52, 61	rexlimddv 3035	1 ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 ifcif 4086 ⟨cop 4183 ∪ cuni 4436 ∩ cint 4475 {copab 4712 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 Oncon0 5723 ℩cio 5849 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 1_𝑜c1o 7553 2_𝑜c2o 7554 +_𝑜 coa 7557 ·_𝑜 comu 7558 ↑_𝑜 coe 7559 CNF ccnf 8558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seqom 7543 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-oexp 7566 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-cnf 8559

This theorem is referenced by: cantnf 8590

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