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Mirrors > Home > MPE Home > Th. List > eliniseg | Structured version Visualization version GIF version |
Description: Membership in the inverse image of a singleton. An application is to express initial segments for an order relation. See for example Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
eliniseg.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
eliniseg | ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliniseg.1 | . 2 ⊢ 𝐶 ∈ V | |
2 | elinisegg 6113 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) | |
3 | 1, 2 | mpan2 691 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2105 Vcvv 3477 {csn 4630 class class class wbr 5147 ◡ccnv 5687 “ cima 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 |
This theorem is referenced by: epin 6115 iniseg 6117 dfco2a 6267 isomin 7356 isoini 7357 fnse 8156 infxpenlem 10050 fpwwe2lem7 10674 fpwwe2lem11 10678 fpwwe2lem12 10679 fpwwe2 10680 canth4 10684 canthwelem 10687 pwfseqlem4 10699 fz1isolem 14496 itg1addlem4 25747 itg1addlem4OLD 25748 elnlfn 31956 pw2f1ocnv 43025 |
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