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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 5932 | . 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 208 ∈ wcel 2110 ∃wrex 3139 Vcvv 3494 class class class wbr 5065 “ cima 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 |
This theorem is referenced by: elima2 5934 rninxp 6035 imaco 6103 isarep1 6441 eliman0 6704 funimass4 6729 isomin 7089 dfsup2 8907 dfac10b 9564 hausmapdom 22107 pi1blem 23642 adjbd1o 29861 imaindm 33022 scutun12 33271 madeval2 33290 brimage 33387 dfrecs2 33411 dfrdg4 33412 dfint3 33413 imagesset 33414 elimaint 39991 elintima 39996 |
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