Theorem bnj207 32889

Description: Technical lemma for bnj852 32929. 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
bnj207.1	⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj207.2	⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑)
bnj207.3	⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓)
bnj207.4	⊢ (𝜒′ ↔ [𝑀 / 𝑛]𝜒)
bnj207.5	⊢ 𝑀 ∈ V

Assertion

Ref	Expression
bnj207	⊢ (𝜒′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′)))

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

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

Proof of Theorem bnj207

Step	Hyp	Ref	Expression
1		bnj207.4	. 2 ⊢ (𝜒′ ↔ [𝑀 / 𝑛]𝜒)
2		bnj207.1	. . . 4 ⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
3	2	sbcbii 3778	. . 3 ⊢ ([𝑀 / 𝑛]𝜒 ↔ [𝑀 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
4		bnj207.5	. . . . 5 ⊢ 𝑀 ∈ V
5		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)
6	5	sbc19.21g 3796	. . . . 5 ⊢ (𝑀 ∈ V → ([𝑀 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
7	4, 6	ax-mp 5	. . . 4 ⊢ ([𝑀 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
8	4	bnj89 32728	. . . . . 6 ⊢ ([𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃!𝑓[𝑀 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
9	4	bnj90 32729	. . . . . . . . 9 ⊢ ([𝑀 / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑀)
10	9	bicomi 223	. . . . . . . 8 ⊢ (𝑓 Fn 𝑀 ↔ [𝑀 / 𝑛]𝑓 Fn 𝑛)
11		bnj207.2	. . . . . . . 8 ⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑)
12		bnj207.3	. . . . . . . 8 ⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓)
13	10, 11, 12, 4	bnj206 32738	. . . . . . 7 ⊢ ([𝑀 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′))
14	13	eubii 2580	. . . . . 6 ⊢ (∃!𝑓[𝑀 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′))
15	8, 14	bitri 274	. . . . 5 ⊢ ([𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′))
16	15	imbi2i 335	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′)))
17	7, 16	bitri 274	. . 3 ⊢ ([𝑀 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′)))
18	3, 17	bitri 274	. 2 ⊢ ([𝑀 / 𝑛]𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′)))
19	1, 18	bitri 274	1 ⊢ (𝜒′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2101  ∃!weu 2563  Vcvv 3434  [wsbc 3718   Fn wfn 6442   FrSe w-bnj15 32699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3436  df-sbc 3719  df-fn 6450

This theorem is referenced by:  bnj600  32927  bnj908  32939