Theorem bnj864 35552

Description: Technical lemma for bnj69 35640. 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
bnj864.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj864.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj864.3	⊢ 𝐷 = (ω ∖ {∅})
bnj864.4	⊢ (𝜒 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷))
bnj864.5	⊢ (𝜃 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Assertion

Ref	Expression
bnj864	⊢ (𝜒 → ∃!𝑓𝜃)

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)   𝜒(𝑦, 𝑓, 𝑖, 𝑛)   𝜃(𝑦, 𝑓, 𝑖, 𝑛)   𝐷(𝑦)   𝑋(𝑦, 𝑖)

Proof of Theorem bnj864

Step	Hyp	Ref	Expression
1		bnj864.1	. . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj864.2	. . . . 5 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj864.3	. . . . 5 ⊢ 𝐷 = (ω ∖ {∅})
4	1, 2, 3	bnj852 35551	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5		df-ral 3077	. . . . . 6 ⊢ (∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∀𝑛(𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
6	5	imbi2i 339	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛(𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
7		19.21v 1971	. . . . 5 ⊢ (∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛(𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
8		impexp 456	. . . . . . 7 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
9		df-3an 1105	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷))
10	9	bicomi 227	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷))
11	10	imbi1i 352	. . . . . . 7 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
12	8, 11	bitr3i 280	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
13	12	albii 1851	. . . . 5 ⊢ (∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
14	6, 7, 13	3bitr2i 302	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
15	4, 14	mpbi 233	. . 3 ⊢ ∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
16	15	spi 2220	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
17		bnj864.4	. 2 ⊢ (𝜒 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷))
18		bnj864.5	. . 3 ⊢ (𝜃 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
19	18	eubii 2610	. 2 ⊢ (∃!𝑓𝜃 ↔ ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
20	16, 17, 19	3imtr4i 295	1 ⊢ (𝜒 → ∃!𝑓𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∃!weu 2593  ∀wral 3076   ∖ cdif 4045  ∅c0 4278  {csn 4583  ∪ ciun 4950  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  ωcom 7877   predc-bnj14 35319   FrSe w-bnj15 35323

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-bnj17 35318  df-bnj14 35320  df-bnj13 35322  df-bnj15 35324

This theorem is used by:  bnj849  35555