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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj96 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 34507. 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.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj96.1 | ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
Ref | Expression |
---|---|
bnj96 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj93 34494 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) | |
2 | dmsnopg 6217 | . . 3 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → dom {〈∅, pred(𝑥, 𝐴, 𝑅)〉} = {∅}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom {〈∅, pred(𝑥, 𝐴, 𝑅)〉} = {∅}) |
4 | bnj96.1 | . . 3 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
5 | 4 | dmeqi 5907 | . 2 ⊢ dom 𝐹 = dom {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
6 | df1o2 8493 | . 2 ⊢ 1o = {∅} | |
7 | 3, 5, 6 | 3eqtr4g 2793 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∅c0 4323 {csn 4629 〈cop 4635 dom cdm 5678 1oc1o 8479 predc-bnj14 34319 FrSe w-bnj15 34323 |
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-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-dm 5688 df-suc 6375 df-1o 8486 df-bnj13 34322 df-bnj15 34324 |
This theorem is referenced by: bnj150 34507 |
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