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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 3185 | . 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | vex 3176 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | eliniseg 5413 | . . 3 ⊢ (𝐵 ∈ V → (𝑥 ∈ (◡𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵)) |
4 | 3 | abbi2dv 2729 | . 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 1475 ∈ wcel 1977 {cab 2596 Vcvv 3173 {csn 4125 class class class wbr 4583 ◡ccnv 5037 “ cima 5041 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: inisegn0 5416 dffr3 5417 dfse2 5418 dfpred2 5606 |
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