Theorem bnj864 33921

Description: Technical lemma for bnj69 34009. 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 33920	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5		df-ral 3062	. . . . . 6 ⊢ (∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∀𝑛(𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
6	5	imbi2i 335	. . . . 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 1942	. . . . 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 451	. . . . . . 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 1089	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷))
10	9	bicomi 223	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷))
11	10	imbi1i 349	. . . . . . 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 276	. . . . . 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 1821	. . . . 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 298	. . . 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 229	. . 3 ⊢ ∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
16	15	spi 2177	. 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 2579	. 2 ⊢ (∃!𝑓𝜃 ↔ ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
20	16, 17, 19	3imtr4i 291	1 ⊢ (𝜒 → ∃!𝑓𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∃!weu 2562  ∀wral 3061   ∖ cdif 3944  ∅c0 4321  {csn 4627  ∪ ciun 4996  suc csuc 6363   Fn wfn 6535  ‘cfv 6540  ωcom 7851   predc-bnj14 33687   FrSe w-bnj15 33691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-reg 9583  ax-inf2 9632

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-bnj17 33686  df-bnj14 33688  df-bnj13 33690  df-bnj15 33692

This theorem is referenced by:  bnj849  33924