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| 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 6082 | . 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 206 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 class class class wbr 5143 “ cima 5688 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 | 
| This theorem is referenced by: elima2 6084 rninxp 6199 imaco 6271 imaindm 6319 isarep1 6656 isarep1OLD 6657 eliman0 6946 funimass4 6973 isomin 7357 dfsup2 9484 dfac10b 10180 hausmapdom 23508 pi1blem 25072 scutun12 27855 madeval2 27892 adjbd1o 32104 brimage 35927 dfrecs2 35951 dfrdg4 35952 dfint3 35953 imagesset 35954 elimaint 43662 elintima 43666 | 
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