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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj97 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
bnj96.1 | ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} |
Ref | Expression |
---|---|
bnj97 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
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)) |
Colors of variables: wff setvar class |
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 |
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