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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 31760. 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 31747 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) | |
2 | 0ex 5109 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | bnj519 31619 | . . . 4 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {〈∅, pred(𝑥, 𝐴, 𝑅)〉}) |
4 | bnj96.1 | . . . . 5 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
5 | 4 | funeqi 6253 | . . . 4 ⊢ (Fun 𝐹 ↔ Fun {〈∅, pred(𝑥, 𝐴, 𝑅)〉}) |
6 | 3, 5 | sylibr 235 | . . 3 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun 𝐹) |
7 | 1, 6 | syl 17 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹) |
8 | opex 5255 | . . . 4 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ V | |
9 | 8 | snid 4512 | . . 3 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
10 | 9, 4 | eleqtrri 2884 | . 2 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ 𝐹 |
11 | funopfv 6592 | . 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 1525 ∈ wcel 2083 Vcvv 3440 ∅c0 4217 {csn 4478 〈cop 4484 Fun wfun 6226 ‘cfv 6232 predc-bnj14 31571 FrSe w-bnj15 31575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-iota 6196 df-fun 6234 df-fv 6240 df-bnj13 31574 df-bnj15 31576 |
This theorem is referenced by: bnj150 31760 |
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