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Mirrors > Home > MPE Home > Th. List > elimasn | Structured version Visualization version GIF version |
Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
elimasn.1 | ⊢ 𝐵 ∈ V |
elimasn.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elimasn | ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimasn.2 | . . 3 ⊢ 𝐶 ∈ V | |
2 | breq2 5069 | . . 3 ⊢ (𝑥 = 𝐶 → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶)) | |
3 | elimasn.1 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | imasng 5950 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥}) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥} |
6 | 1, 2, 5 | elab2 3669 | . 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶) |
7 | df-br 5066 | . 2 ⊢ (𝐵𝐴𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐴) | |
8 | 6, 7 | bitri 277 | 1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 {cab 2799 Vcvv 3494 {csn 4566 〈cop 4572 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-sbc 3772 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: elimasng 5954 dfco2 6097 dfco2a 6098 ressn 6135 funfvima3 6997 frxp 7819 marypha1lem 8896 gsum2dlem1 19089 gsum2dlem2 19090 gsum2d 19091 gsum2d2 19093 ovoliunlem1 24102 iunsnima 30368 dfcnv2 30421 gsummpt2co 30686 gsummpt2d 30687 dmscut 33272 scutf 33273 funpartfun 33404 areaquad 39821 dffrege76 40283 frege97 40304 frege98 40305 frege109 40316 frege110 40317 frege131 40338 frege133 40340 |
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