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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 34721. 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 34708 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) | |
2 | 0ex 5312 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | bnj519 34581 | . . . 4 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {〈∅, pred(𝑥, 𝐴, 𝑅)〉}) |
4 | bnj96.1 | . . . . 5 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
5 | 4 | funeqi 6580 | . . . 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 5470 | . . . 4 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ V | |
9 | 8 | snid 4669 | . . 3 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
10 | 9, 4 | eleqtrri 2825 | . 2 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ 𝐹 |
11 | funopfv 6953 | . 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 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 {csn 4633 〈cop 4639 Fun wfun 6548 ‘cfv 6554 predc-bnj14 34533 FrSe w-bnj15 34537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-bnj13 34536 df-bnj15 34538 |
This theorem is referenced by: bnj150 34721 |
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