Theorem bnj130 30198

Description: Technical lemma for bnj151 30201. 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 3458	. 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 30044	. . . . . . . . . 10 ⊢ 1𝑜 ∈ V
5	4	bnj90 30042	. . . . . . . . 9 ⊢ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 1𝑜)
6	5	bicomi 213	. . . . . . . 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 1244	. . . . . . 7 ⊢ ((𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ∧ [1𝑜 / 𝑛]𝜑 ∧ [1𝑜 / 𝑛]𝜓))
10		sbc3an 3461	. . . . . . 7 ⊢ ([1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ∧ [1𝑜 / 𝑛]𝜑 ∧ [1𝑜 / 𝑛]𝜓))
11	9, 10	bitr4i 266	. . . . . 6 ⊢ ((𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ [1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
12	11	eubii 2480	. . . . 5 ⊢ (∃!𝑓(𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ ∃!𝑓[1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
13	4	bnj89 30041	. . . . 5 ⊢ ([1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃!𝑓[1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
14	12, 13	bitr4i 266	. . . 4 ⊢ (∃!𝑓(𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ [1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
15	14	imbi2i 325	. . 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 1830	. . . . 5 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)
17	16	sbc19.21g 3469	. . . 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 266	. 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 291	1 ⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∃!weu 2458  Vcvv 3173  [wsbc 3402   Fn wfn 5799  1_𝑜c1o 7440   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

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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-suc 5646  df-fn 5807  df-1o 7447

This theorem is referenced by:  bnj151  30201