Theorem bnj601 32194

Description: Technical lemma for bnj852 32195. 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 263	. 2 ⊢ ([𝑚 / 𝑛]𝜑 ↔ [𝑚 / 𝑛]𝜑)
7		biid 263	. 2 ⊢ ([𝑚 / 𝑛]𝜓 ↔ [𝑚 / 𝑛]𝜓)
8		biid 263	. 2 ⊢ ([𝑚 / 𝑛]𝜒 ↔ [𝑚 / 𝑛]𝜒)
9		bnj602 32189	. . . . . . 7 ⊢ (𝑦 = 𝑧 → pred(𝑦, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
10	9	cbviunv 4967	. . . . . 6 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)
11	10	opeq2i 4809	. . . . 5 ⊢ ⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩ = ⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩
12	11	sneqi 4580	. . . 4 ⊢ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}
13	12	uneq2i 4138	. . 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 2832	. 2 ⊢ (𝑓 ∪ {⟨𝑚, ∪ 𝑧 ∈ (𝑓‘𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
21		biid 263	. 2 ⊢ ((𝑓 Fn 𝑚 ∧ [𝑚 / 𝑛]𝜑 ∧ [𝑚 / 𝑛]𝜓) ↔ (𝑓 Fn 𝑚 ∧ [𝑚 / 𝑛]𝜑 ∧ [𝑚 / 𝑛]𝜓))
22		biid 263	. 2 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
23		biid 263	. 2 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
24		biid 263	. 2 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
25		biid 263	. 2 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
26		eqid 2823	. 2 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
27		eqid 2823	. 2 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
28		eqid 2823	. 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 2823	. 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 32193	1 ⊢ (𝑛 ≠ 1o → ((𝑛 ∈ 𝐷 ∧ 𝜃) → 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃!weu 2653   ≠ wne 3018  ∀wral 3140  [wsbc 3774   ∖ cdif 3935   ∪ cun 3936  ∅c0 4293  {csn 4569  ⟨cop 4575  ∪ ciun 4921   class class class wbr 5068   E cep 5466  suc csuc 6195   Fn wfn 6352  ‘cfv 6357  ωcom 7582  1_oc1o 8097   ∧ w-bnj17 31958   predc-bnj14 31960   FrSe w-bnj15 31964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-reg 9058

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-1o 8104  df-bnj17 31959  df-bnj14 31961  df-bnj13 31963  df-bnj15 31965

This theorem is referenced by:  bnj852  32195