Theorem bnj97 34849

Description: Technical lemma for bnj150 34859. 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 34846	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2		0ex 5246	. . . . 5 ⊢ ∅ ∈ V
3	2	bnj519 34719	. . . 4 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
4		bnj96.1	. . . . 5 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
5	4	funeqi 6503	. . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
6	3, 5	sylibr 234	. . 3 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun 𝐹)
7	1, 6	syl 17	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹)
8		opex 5407	. . . 4 ⊢ ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ V
9	8	snid 4614	. . 3 ⊢ ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
10	9, 4	eleqtrri 2827	. 2 ⊢ ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ 𝐹
11		funopfv 6872	. 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 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284  {csn 4577  ⟨cop 4583  Fun wfun 6476  ‘cfv 6482   predc-bnj14 34671   FrSe w-bnj15 34675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-bnj13 34674  df-bnj15 34676

This theorem is referenced by:  bnj150  34859