Theorem bnj97 33865

Description: Technical lemma for bnj150 33875. 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.)

Hypothesis

Ref	Expression
bnj96.1	⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}

Assertion

Ref	Expression
bnj97	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem bnj97

Step	Hyp	Ref	Expression
1		bnj93 33862	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2		0ex 5306	. . . . 5 ⊢ ∅ ∈ V
3	2	bnj519 33735	. . . 4 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
4		bnj96.1	. . . . 5 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
5	4	funeqi 6566	. . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
6	3, 5	sylibr 233	. . 3 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun 𝐹)
7	1, 6	syl 17	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹)
8		opex 5463	. . . 4 ⊢ ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ V
9	8	snid 4663	. . 3 ⊢ ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
10	9, 4	eleqtrri 2832	. 2 ⊢ ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ 𝐹
11		funopfv 6940	. 2 ⊢ (Fun 𝐹 → (⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ 𝐹 → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)))
12	7, 10, 11	mpisyl 21	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∅c0 4321  {csn 4627  ⟨cop 4633  Fun wfun 6534  ‘cfv 6540   predc-bnj14 33687   FrSe w-bnj15 33691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-bnj13 33690  df-bnj15 33692

This theorem is referenced by:  bnj150  33875