Theorem bnj601 35416

Description: Technical lemma for bnj852 35417. 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
bnj601.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj601.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj601.3	⊢ 𝐷 = (ω ∖ {∅})
bnj601.4	⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj601.5	⊢ (𝜃 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒))

Assertion

Ref	Expression
bnj601	⊢ (𝑛 ≠ 1o → ((𝑛 ∈ 𝐷 ∧ 𝜃) → 𝜒))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑥,𝑓,𝑚,𝑛   𝜑,𝑖,𝑚   𝜓,𝑚

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑓, 𝑛)   𝜓(𝑥, 𝑦, 𝑓, 𝑖, 𝑛)   𝜒(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝜃(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝐴(𝑥)   𝐷(𝑥, 𝑦, 𝑚, 𝑛)   𝑅(𝑥)

Proof of Theorem bnj601

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

Step	Hyp	Ref	Expression
1		bnj601.1	. 2 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2		bnj601.2	. 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj601.3	. 2 ⊢ 𝐷 = (ω ∖ {∅})
4		bnj601.4	. 2 ⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
5		bnj601.5	. 2 ⊢ (𝜃 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒))
6		biid 264	. 2 ⊢ ([𝑚 / 𝑛]𝜑 ↔ [𝑚 / 𝑛]𝜑)
7		biid 264	. 2 ⊢ ([𝑚 / 𝑛]𝜓 ↔ [𝑚 / 𝑛]𝜓)
8		biid 264	. 2 ⊢ ([𝑚 / 𝑛]𝜒 ↔ [𝑚 / 𝑛]𝜒)
9		bnj602 35411	. . . . . . 7 ⊢ (𝑦 = 𝑧 → pred(𝑦, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
10	9	cbviunv 5001	. . . . . 6 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)
11	10	opeq2i 4840	. . . . 5 ⊢ ⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩ = ⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩
12	11	sneqi 4598	. . . 4 ⊢ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}
13	12	uneq2i 4115	. . 3 ⊢ (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩})
14		dfsbcq 3744	. . 3 ⊢ ((𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) → ([(𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜑 ↔ [(𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜑))
15	13, 14	ax-mp 5	. 2 ⊢ ([(𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜑 ↔ [(𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜑)
16		dfsbcq 3744	. . 3 ⊢ ((𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) → ([(𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜓 ↔ [(𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜓))
17	13, 16	ax-mp 5	. 2 ⊢ ([(𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜓 ↔ [(𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜓)
18		dfsbcq 3744	. . 3 ⊢ ((𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) → ([(𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜒 ↔ [(𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜒))
19	13, 18	ax-mp 5	. 2 ⊢ ([(𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜒 ↔ [(𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜒)
20	13	eqcomi 2771	. 2 ⊢ (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
21		biid 264	. 2 ⊢ ((𝑓 Fn 𝑚 ∧ [𝑚 / 𝑛]𝜑 ∧ [𝑚 / 𝑛]𝜓) ↔ (𝑓 Fn 𝑚 ∧ [𝑚 / 𝑛]𝜑 ∧ [𝑚 / 𝑛]𝜓))
22		biid 264	. 2 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
23		biid 264	. 2 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
24		biid 264	. 2 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
25		biid 264	. 2 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
26		eqid 2762	. 2 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
27		eqid 2762	. 2 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
28		eqid 2762	. 2 ⊢ ∪ 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑖) pred(𝑦, 𝐴, 𝑅)
29		eqid 2762	. 2 ⊢ ∪ 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑝) pred(𝑦, 𝐴, 𝑅)
30	1, 2, 3, 4, 5, 6, 7, 8, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 20	bnj600 35415	1 ⊢ (𝑛 ≠ 1o → ((𝑛 ∈ 𝐷 ∧ 𝜃) → 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃!weu 2595   ≠ wne 2957  ∀wral 3078  [wsbc 3742   ∖ cdif 3899   ∪ cun 3900  ∅c0 4282  {csn 4587  ⟨cop 4593  ∪ ciun 4954   class class class wbr 5107   E cep 5558  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7865  1_oc1o 8451   ∧ w-bnj17 35183   predc-bnj14 35185   FrSe w-bnj15 35189

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739  ax-reg 9567

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-om 7866  df-1o 8458  df-bnj17 35184  df-bnj14 35186  df-bnj13 35188  df-bnj15 35190

This theorem is used by:  bnj852  35417