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Mirrors > Home > MPE Home > Th. List > iniseg | Structured version Visualization version GIF version |
Description: An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) |
Ref | Expression |
---|---|
iniseg | ⊢ (𝐵 ∈ 𝑉 → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3461 | . 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | vex 3447 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | eliniseg 6044 | . . 3 ⊢ (𝐵 ∈ V → (𝑥 ∈ (◡𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵)) |
4 | 3 | abbi2dv 2876 | . 2 ⊢ (𝐵 ∈ V → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵}) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐵 ∈ 𝑉 → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {cab 2714 Vcvv 3443 {csn 4584 class class class wbr 5103 ◡ccnv 5630 “ cima 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-xp 5637 df-cnv 5639 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 |
This theorem is referenced by: inisegn0 6048 dffr3 6049 dfse2 6050 dfpred2 6261 predres 6291 |
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