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Mirrors > Home > MPE Home > Th. List > elima | Structured version Visualization version GIF version |
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.) |
Ref | Expression |
---|---|
elima.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elima | ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elima.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elimag 6061 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2106 ∃wrex 3070 Vcvv 3474 class class class wbr 5147 “ cima 5678 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 |
This theorem is referenced by: elima2 6063 rninxp 6175 imaco 6247 imaindm 6295 isarep1 6634 isarep1OLD 6635 eliman0 6928 funimass4 6953 isomin 7330 dfsup2 9435 dfac10b 10130 hausmapdom 22995 pi1blem 24546 scutun12 27300 madeval2 27337 adjbd1o 31325 brimage 34886 dfrecs2 34910 dfrdg4 34911 dfint3 34912 imagesset 34913 elimaint 42385 elintima 42389 |
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