Theorem bnj97 34844

Description: Technical lemma for bnj150 34854. 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 34841	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2		0ex 5325	. . . . 5 ⊢ ∅ ∈ V
3	2	bnj519 34714	. . . 4 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
4		bnj96.1	. . . . 5 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
5	4	funeqi 6601	. . . 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 5484	. . . 4 ⊢ ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ V
9	8	snid 4684	. . 3 ⊢ ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
10	9, 4	eleqtrri 2843	. 2 ⊢ ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ 𝐹
11		funopfv 6974	. 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 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  {csn 4648  ⟨cop 4654  Fun wfun 6569  ‘cfv 6575   predc-bnj14 34666   FrSe w-bnj15 34670

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 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6527  df-fun 6577  df-fv 6583  df-bnj13 34669  df-bnj15 34671

This theorem is referenced by:  bnj150  34854