![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj97 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 31284. 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 31271 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) | |
2 | 0ex 4924 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | bnj519 31142 | . . . 4 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {〈∅, pred(𝑥, 𝐴, 𝑅)〉}) |
4 | bnj96.1 | . . . . 5 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
5 | 4 | funeqi 6052 | . . . 4 ⊢ (Fun 𝐹 ↔ Fun {〈∅, pred(𝑥, 𝐴, 𝑅)〉}) |
6 | 3, 5 | sylibr 224 | . . 3 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun 𝐹) |
7 | 1, 6 | syl 17 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹) |
8 | opex 5060 | . . . 4 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ V | |
9 | 8 | snid 4347 | . . 3 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
10 | 9, 4 | eleqtrri 2849 | . 2 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ 𝐹 |
11 | funopfv 6376 | . 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 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∅c0 4063 {csn 4316 〈cop 4322 Fun wfun 6025 ‘cfv 6031 predc-bnj14 31094 FrSe w-bnj15 31098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-iota 5994 df-fun 6033 df-fv 6039 df-bnj13 31097 df-bnj15 31099 |
This theorem is referenced by: bnj150 31284 |
Copyright terms: Public domain | W3C validator |