Theorem bnj601 33826

Description: Technical lemma for bnj852 33827. 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 260	. 2 ⊢ ([𝑚 / 𝑛]𝜑 ↔ [𝑚 / 𝑛]𝜑)
7		biid 260	. 2 ⊢ ([𝑚 / 𝑛]𝜓 ↔ [𝑚 / 𝑛]𝜓)
8		biid 260	. 2 ⊢ ([𝑚 / 𝑛]𝜒 ↔ [𝑚 / 𝑛]𝜒)
9		bnj602 33821	. . . . . . 7 ⊢ (𝑦 = 𝑧 → pred(𝑦, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
10	9	cbviunv 5037	. . . . . 6 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)
11	10	opeq2i 4871	. . . . 5 ⊢ ⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩ = ⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩
12	11	sneqi 4634	. . . 4 ⊢ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}
13	12	uneq2i 4157	. . 3 ⊢ (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩})
14		dfsbcq 3776	. . 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 3776	. . 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 3776	. . 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 2741	. 2 ⊢ (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
21		biid 260	. 2 ⊢ ((𝑓 Fn 𝑚 ∧ [𝑚 / 𝑛]𝜑 ∧ [𝑚 / 𝑛]𝜓) ↔ (𝑓 Fn 𝑚 ∧ [𝑚 / 𝑛]𝜑 ∧ [𝑚 / 𝑛]𝜓))
22		biid 260	. 2 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
23		biid 260	. 2 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
24		biid 260	. 2 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
25		biid 260	. 2 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
26		eqid 2732	. 2 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
27		eqid 2732	. 2 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
28		eqid 2732	. 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 2732	. 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 33825	1 ⊢ (𝑛 ≠ 1o → ((𝑛 ∈ 𝐷 ∧ 𝜃) → 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃!weu 2562   ≠ wne 2940  ∀wral 3061  [wsbc 3774   ∖ cdif 3942   ∪ cun 3943  ∅c0 4319  {csn 4623  ⟨cop 4629  ∪ ciun 4991   class class class wbr 5142   E cep 5573  suc csuc 6356   Fn wfn 6528  ‘cfv 6533  ωcom 7839  1_oc1o 8443   ∧ w-bnj17 33592   predc-bnj14 33594   FrSe w-bnj15 33598

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 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709  ax-reg 9571

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-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-fv 6541  df-om 7840  df-1o 8450  df-bnj17 33593  df-bnj14 33595  df-bnj13 33597  df-bnj15 33599

This theorem is referenced by:  bnj852  33827