Theorem bnj130 32148

Description: Technical lemma for bnj151 32151. 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
bnj130.1	⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj130.2	⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
bnj130.3	⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)
bnj130.4	⊢ (𝜃′ ↔ [1o / 𝑛]𝜃)

Assertion

Ref	Expression
bnj130	⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))

Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑓,𝑛   𝑥,𝑛

Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝜓(𝑥,𝑓,𝑛)   𝜃(𝑥,𝑓,𝑛)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝜑′(𝑥,𝑓,𝑛)   𝜓′(𝑥,𝑓,𝑛)   𝜃′(𝑥,𝑓,𝑛)

Proof of Theorem bnj130

Step	Hyp	Ref	Expression
1		bnj130.1	. . 3 ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
2	1	sbcbii 3831	. 2 ⊢ ([1o / 𝑛]𝜃 ↔ [1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
3		bnj130.4	. 2 ⊢ (𝜃′ ↔ [1o / 𝑛]𝜃)
4		bnj105 31996	. . . . . . . . . 10 ⊢ 1o ∈ V
5	4	bnj90 31994	. . . . . . . . 9 ⊢ ([1o / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 1o)
6	5	bicomi 226	. . . . . . . 8 ⊢ (𝑓 Fn 1o ↔ [1o / 𝑛]𝑓 Fn 𝑛)
7		bnj130.2	. . . . . . . 8 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
8		bnj130.3	. . . . . . . 8 ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)
9	6, 7, 8	3anbi123i 1151	. . . . . . 7 ⊢ ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ ([1o / 𝑛]𝑓 Fn 𝑛 ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓))
10		sbc3an 3840	. . . . . . 7 ⊢ ([1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([1o / 𝑛]𝑓 Fn 𝑛 ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓))
11	9, 10	bitr4i 280	. . . . . 6 ⊢ ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
12	11	eubii 2670	. . . . 5 ⊢ (∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ ∃!𝑓[1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
13	4	bnj89 31993	. . . . 5 ⊢ ([1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃!𝑓[1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
14	12, 13	bitr4i 280	. . . 4 ⊢ (∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ [1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
15	14	imbi2i 338	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
16		nfv 1915	. . . . 5 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)
17	16	sbc19.21g 3848	. . . 4 ⊢ (1o ∈ V → ([1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
18	4, 17	ax-mp 5	. . 3 ⊢ ([1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [1o / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
19	15, 18	bitr4i 280	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ [1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
20	2, 3, 19	3bitr4i 305	1 ⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114  ∃!weu 2653  Vcvv 3496  [wsbc 3774   Fn wfn 6352  1_oc1o 8097   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-sep 5205  ax-nul 5212  ax-pow 5268

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  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-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-pw 4543  df-sn 4570  df-suc 6199  df-fn 6360  df-1o 8104

This theorem is referenced by:  bnj151  32151