Theorem bnj864 32902

Description: Technical lemma for bnj69 32990. 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 32901	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5		df-ral 3069	. . . . . 6 ⊢ (∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∀𝑛(𝑛 ∈ 𝐷 → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
6	5	imbi2i 336	. . . . 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 1088	. . . . . . . . 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 350	. . . . . . 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 1822	. . . . 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 299	. . . 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 2585	. 2 ⊢ (∃!𝑓𝜃 ↔ ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
20	16, 17, 19	3imtr4i 292	1 ⊢ (𝜒 → ∃!𝑓𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  ∀wral 3064   ∖ cdif 3884  ∅c0 4256  {csn 4561  ∪ ciun 4924  suc csuc 6268   Fn wfn 6428  ‘cfv 6433  ωcom 7712   predc-bnj14 32667   FrSe w-bnj15 32671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-bnj17 32666  df-bnj14 32668  df-bnj13 32670  df-bnj15 32672

This theorem is referenced by:  bnj849  32905