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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 3465 | . 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | vex 3451 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | eliniseg 6050 | . . 3 ⊢ (𝐵 ∈ V → (𝑥 ∈ (◡𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵)) |
4 | 3 | abbi2dv 2868 | . 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 1542 ∈ wcel 2107 {cab 2710 Vcvv 3447 {csn 4590 class class class wbr 5109 ◡ccnv 5636 “ cima 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 |
This theorem is referenced by: inisegn0 6054 dffr3 6055 dfse2 6056 dfpred2 6267 predres 6297 |
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