Theorem bnj852 35279

Description: Technical lemma for bnj69 35368. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj852.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj852.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj852.3	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj852	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑛

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝐷(𝑦)   𝑋(𝑦,𝑖)

Proof of Theorem bnj852

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2852	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 = 𝑋)
2	1	adantl 486	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 𝑥 = 𝑋)
3	2	ancri 558	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (∃𝑥 𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)))
4	3	bnj534 35098	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑥(𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)))
5		eleq1 2858	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
6	5	anbi2d 641	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)))
7	6	biimpar 482	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴))
8		biid 264	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐷 (𝑧 E 𝑛 → [𝑧 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))) ↔ ∀𝑧 ∈ 𝐷 (𝑧 E 𝑛 → [𝑧 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))))
9		bnj852.3	. . . . . . . . 9 ⊢ 𝐷 = (ω ∖ {∅})
10		omex 9615	. . . . . . . . . 10 ⊢ ω ∈ V
11		difexg 5303	. . . . . . . . . 10 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (ω ∖ {∅}) ∈ V
13	9, 12	eqeltri 2866	. . . . . . . 8 ⊢ 𝐷 ∈ V
14		zfregfr 9576	. . . . . . . 8 ⊢ E Fr 𝐷
15	8, 13, 14	bnj157 35217	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝐷 (∀𝑧 ∈ 𝐷 (𝑧 E 𝑛 → [𝑧 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))) → ∀𝑛 ∈ 𝐷 ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))
16		biid 264	. . . . . . . . . 10 ⊢ ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
17		bnj852.2	. . . . . . . . . 10 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
18		biid 264	. . . . . . . . . 10 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))
19	16, 17, 9, 18, 8	bnj153 35238	. . . . . . . . 9 ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ ∀𝑧 ∈ 𝐷 (𝑧 E 𝑛 → [𝑧 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))))
20	16, 17, 9, 18, 8	bnj601 35278	. . . . . . . . 9 ⊢ (𝑛 ≠ 1o → ((𝑛 ∈ 𝐷 ∧ ∀𝑧 ∈ 𝐷 (𝑧 E 𝑛 → [𝑧 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))))
21	19, 20	pm2.61ine 3048	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ ∀𝑧 ∈ 𝐷 (𝑧 E 𝑛 → [𝑧 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))
22	21	ex 417	. . . . . . 7 ⊢ (𝑛 ∈ 𝐷 → (∀𝑧 ∈ 𝐷 (𝑧 E 𝑛 → [𝑧 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))))
23	15, 22	mprg 3092	. . . . . 6 ⊢ ∀𝑛 ∈ 𝐷 ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))
24		r19.21v 3197	. . . . . 6 ⊢ (∀𝑛 ∈ 𝐷 ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))
25	23, 24	mpbi 233	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))
26	7, 25	syl 18	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))
27		bnj602 35273	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
28	27	eqeq2d 2781	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
29		bnj852.1	. . . . . . . . 9 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
30	28, 29	bitr4di 292	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ 𝜑))
31	30	3anbi2d 1467	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
32	31	eubidv 2621	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
33	32	ralbidv 3195	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
34	33	adantr 485	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) → (∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
35	26, 34	mpbid 235	. . 3 ⊢ ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
36	4, 35	bnj593 35104	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑥∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
37	36	bnj937 35130	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568  ∃wex 1807   ∈ wcel 2150  ∃!weu 2603  ∀wral 3086  Vcvv 3462  [wsbc 3752   ∖ cdif 3910  ∅c0 4294  {csn 4594  ∪ ciun 4961   class class class wbr 5114   E cep 5564  suc csuc 6366   Fn wfn 6535  ‘cfv 6540  ωcom 7865  1_oc1o 8449   predc-bnj14 35047   FrSe w-bnj15 35051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736  ax-reg 9557  ax-inf2 9613

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7866  df-1o 8456  df-bnj17 35046  df-bnj14 35048  df-bnj13 35050  df-bnj15 35052

This theorem is referenced by:  bnj864  35280  bnj865  35281  bnj906  35288