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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj96 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 30200. 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 30187 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) | |
2 | dmsnopg 5524 | . . 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 5247 | . 2 ⊢ dom 𝐹 = dom {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
6 | df1o2 7459 | . 2 ⊢ 1𝑜 = {∅} | |
7 | 3, 5, 6 | 3eqtr4g 2669 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 〈cop 4131 dom cdm 5038 1𝑜c1o 7440 predc-bnj14 30007 FrSe w-bnj15 30011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 df-suc 5646 df-1o 7447 df-bnj13 30010 df-bnj15 30012 |
This theorem is referenced by: bnj150 30200 |
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