bnj151 - Mathbox for Jonathan Ben-Naim

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Theorem bnj151 30947

Description: Technical lemma for bnj153 30950. 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 30946	. . . . . 6 ⊢ 𝜃₀
13	12, 6	mpbi 220	. . . . 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 30939	. . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
20	14, 15, 16, 17, 18, 19	bnj149 30945	. . . . . 6 ⊢ 𝜃₁
21	20, 14	mpbi 220	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′))
22		eu5 2496	. . . . 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 698	. . . 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 30944	. . . 4 ⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1_𝑜 ∧ 𝜑′ ∧ 𝜓′)))
27	23, 26	mpbir 221	. . 3 ⊢ 𝜃′
28		sbceq1a 3446	. . . 4 ⊢ (𝑛 = 1_𝑜 → (𝜃 ↔ [1_𝑜 / 𝑛]𝜃))
29	28, 25	syl6bbr 278	. . 3 ⊢ (𝑛 = 1_𝑜 → (𝜃 ↔ 𝜃′))
30	27, 29	mpbiri 248	. 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 ∃*wmo 2471 ∀wral 2912 [wsbc 3435 ∖ cdif 3571 ∅c0 3915 {csn 4177 ⟨cop 4183 ∪ ciun 4520 class class class wbr 4653 E cep 5028 suc csuc 5725 Fn wfn 5883 ‘cfv 5888 ωcom 7065 1_𝑜c1o 7553 predc-bnj14 30754 FrSe w-bnj15 30758

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-1o 7560 df-bnj13 30757 df-bnj15 30759

This theorem is referenced by: bnj153 30950

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