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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj96 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj150 34890. 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 34877 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) | |
| 2 | dmsnopg 6233 | . . 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 5915 | . 2 ⊢ dom 𝐹 = dom {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
| 6 | df1o2 8513 | . 2 ⊢ 1o = {∅} | |
| 7 | 3, 5, 6 | 3eqtr4g 2802 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 〈cop 4632 dom cdm 5685 1oc1o 8499 predc-bnj14 34702 FrSe w-bnj15 34706 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 df-suc 6390 df-1o 8506 df-bnj13 34705 df-bnj15 34707 |
| This theorem is referenced by: bnj150 34890 |
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