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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj96 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 32150. 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 32137 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) | |
2 | dmsnopg 6072 | . . 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 5775 | . 2 ⊢ dom 𝐹 = dom {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
6 | df1o2 8118 | . 2 ⊢ 1o = {∅} | |
7 | 3, 5, 6 | 3eqtr4g 2883 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 {csn 4569 〈cop 4575 dom cdm 5557 1oc1o 8097 predc-bnj14 31960 FrSe w-bnj15 31964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-dm 5567 df-suc 6199 df-1o 8104 df-bnj13 31963 df-bnj15 31965 |
This theorem is referenced by: bnj150 32150 |
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