bnj151 - Mathbox for Jonathan Ben-Naim

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Theorem bnj151 30201

Description: Technical lemma for bnj153 30204. 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
bnj151.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj151.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj151.3	⊢ 𝐷 = (ω ∖ {∅})
bnj151.4	⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj151.5	⊢ (𝜏 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜃))
bnj151.6	⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj151.7	⊢ (𝜑′ ↔ [1_𝑜 / 𝑛]𝜑)
bnj151.8	⊢ (𝜓′ ↔ [1_𝑜 / 𝑛]𝜓)
bnj151.9	⊢ (𝜃′ ↔ [1_𝑜 / 𝑛]𝜃)
bnj151.10	⊢ (𝜃₀ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′)))
bnj151.11	⊢ (𝜃₁ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′)))
bnj151.12	⊢ (𝜁′ ↔ [1_𝑜 / 𝑛]𝜁)
bnj151.13	⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
bnj151.14	⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
bnj151.15	⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
bnj151.16	⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)
bnj151.17	⊢ (𝜁₀ ↔ (𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′))
bnj151.18	⊢ (𝜁₁ ↔ [𝑔 / 𝑓]𝜁₀)
bnj151.19	⊢ (𝜑₁ ↔ [𝑔 / 𝑓]𝜑′)
bnj151.20	⊢ (𝜓₁ ↔ [𝑔 / 𝑓]𝜓′)

**Assertion**
Ref	Expression
bnj151	⊢ (𝑛 = 1_𝑜 → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

Distinct variable groups: 𝐴,𝑓,𝑔,𝑥 𝐴,𝑛,𝑓,𝑥 𝑓,𝐹,𝑖,𝑦 𝑅,𝑓,𝑔,𝑥 𝑅,𝑛 𝑓,𝜁₁ 𝑓,𝜁″ 𝑔,𝜁₀ 𝑖,𝑛,𝑦 𝑚,𝑛 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜓(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜃(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜏(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜁(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝐴(𝑦,𝑖,𝑚) 𝐷(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝑅(𝑦,𝑖,𝑚) 𝐹(𝑥,𝑔,𝑚,𝑛) 𝜑′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜓′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜃′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜁′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜑″(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜓″(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜁″(𝑥,𝑦,𝑔,𝑖,𝑚,𝑛) 𝜃₀(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜁₀(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛) 𝜑₁(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜓₁(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜃₁(𝑥,𝑦,𝑓,𝑔,𝑖,𝑚,𝑛) 𝜁₁(𝑥,𝑦,𝑔,𝑖,𝑚,𝑛)

**Proof of Theorem bnj151**
Step	Hyp	Ref	Expression
1		bnj151.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2		bnj151.2	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj151.6	. . . . . . 7 ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
4		bnj151.7	. . . . . . 7 ⊢ (𝜑′ ↔ [1_𝑜 / 𝑛]𝜑)
5		bnj151.8	. . . . . . 7 ⊢ (𝜓′ ↔ [1_𝑜 / 𝑛]𝜓)
6		bnj151.10	. . . . . . 7 ⊢ (𝜃₀ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′)))
7		bnj151.12	. . . . . . 7 ⊢ (𝜁′ ↔ [1_𝑜 / 𝑛]𝜁)
8		bnj151.13	. . . . . . 7 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
9		bnj151.14	. . . . . . 7 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
10		bnj151.15	. . . . . . 7 ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
11		bnj151.16	. . . . . . 7 ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	bnj150 30200	. . . . . 6 ⊢ 𝜃₀
13	12, 6	mpbi 219	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′))
14		bnj151.11	. . . . . . 7 ⊢ (𝜃₁ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′)))
15		bnj151.17	. . . . . . 7 ⊢ (𝜁₀ ↔ (𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′))
16		bnj151.18	. . . . . . 7 ⊢ (𝜁₁ ↔ [𝑔 / 𝑓]𝜁₀)
17		bnj151.19	. . . . . . 7 ⊢ (𝜑₁ ↔ [𝑔 / 𝑓]𝜑′)
18		bnj151.20	. . . . . . 7 ⊢ (𝜓₁ ↔ [𝑔 / 𝑓]𝜓′)
19	1, 4	bnj118 30193	. . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
20	14, 15, 16, 17, 18, 19	bnj149 30199	. . . . . 6 ⊢ 𝜃₁
21	20, 14	mpbi 219	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′))
22		eu5 2484	. . . . 5 ⊢ (∃!𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ (∃𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′) ∧ ∃*𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′)))
23	13, 21, 22	sylanbrc 695	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′))
24		bnj151.4	. . . . 5 ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
25		bnj151.9	. . . . 5 ⊢ (𝜃′ ↔ [1_𝑜 / 𝑛]𝜃)
26	24, 4, 5, 25	bnj130 30198	. . . 4 ⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′)))
27	23, 26	mpbir 220	. . 3 ⊢ 𝜃′
28		sbceq1a 3413	. . . 4 ⊢ (𝑛 = 1_𝑜 → (𝜃 ↔ [1_𝑜 / 𝑛]𝜃))
29	28, 25	syl6bbr 277	. . 3 ⊢ (𝑛 = 1_𝑜 → (𝜃 ↔ 𝜃′))
30	27, 29	mpbiri 247	. 2 ⊢ (𝑛 = 1_𝑜 → 𝜃)
31	30	a1d 25	1 ⊢ (𝑛 = 1_𝑜 → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃!weu 2458 ∃*wmo 2459 ∀wral 2896 [wsbc 3402 ∖ cdif 3537 ∅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 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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 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-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 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-1o 7447 df-bnj13 30010 df-bnj15 30012

This theorem is referenced by: bnj153 30204

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