Theorem bnj601 30244

Description: Technical lemma for bnj852 30245. 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 250	. 2 ⊢ ([𝑚 / 𝑛]𝜑 ↔ [𝑚 / 𝑛]𝜑)
7		biid 250	. 2 ⊢ ([𝑚 / 𝑛]𝜓 ↔ [𝑚 / 𝑛]𝜓)
8		biid 250	. 2 ⊢ ([𝑚 / 𝑛]𝜒 ↔ [𝑚 / 𝑛]𝜒)
9		bnj602 30239	. . . . . . 7 ⊢ (𝑦 = 𝑧 → pred(𝑦, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
10	9	cbviunv 4495	. . . . . 6 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)
11	10	opeq2i 4344	. . . . 5 ⊢ ⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩ = ⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩
12	11	sneqi 4136	. . . 4 ⊢ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}
13	12	uneq2i 3726	. . 3 ⊢ (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩})
14		dfsbcq 3404	. . 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 3404	. . 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 3404	. . 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 2619	. 2 ⊢ (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
21		biid 250	. 2 ⊢ ((𝑓 Fn 𝑚 ∧ [𝑚 / 𝑛]𝜑 ∧ [𝑚 / 𝑛]𝜓) ↔ (𝑓 Fn 𝑚 ∧ [𝑚 / 𝑛]𝜑 ∧ [𝑚 / 𝑛]𝜓))
22		biid 250	. 2 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
23		biid 250	. 2 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
24		biid 250	. 2 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
25		biid 250	. 2 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
26		eqid 2610	. 2 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
27		eqid 2610	. 2 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
28		eqid 2610	. 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 2610	. 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 30243	1 ⊢ (𝑛 ≠ 1𝑜 → ((𝑛 ∈ 𝐷 ∧ 𝜃) → 𝜒))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  ∀wral 2896  [wsbc 3402   ∖ cdif 3537   ∪ cun 3538  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583   E cep 4947  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957  1_𝑜c1o 7440   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012

This theorem is referenced by:  bnj852  30245