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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 5072 | . . 3 ⊢ (𝑥 = 𝐶 → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶)) | |
3 | elimasn.1 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | imasng 5953 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥}) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥} |
6 | 1, 2, 5 | elab2 3672 | . 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶) |
7 | df-br 5069 | . 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 1537 ∈ wcel 2114 {cab 2801 Vcvv 3496 {csn 4569 〈cop 4575 class class class wbr 5068 “ cima 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 |
This theorem is referenced by: elimasng 5957 dfco2 6100 dfco2a 6101 ressn 6138 funfvima3 7000 frxp 7822 marypha1lem 8899 gsum2dlem1 19092 gsum2dlem2 19093 gsum2d 19094 gsum2d2 19096 ovoliunlem1 24105 iunsnima 30371 dfcnv2 30424 gsummpt2co 30688 gsummpt2d 30689 dmscut 33274 scutf 33275 funpartfun 33406 areaquad 39830 dffrege76 40292 frege97 40313 frege98 40314 frege109 40325 frege110 40326 frege131 40347 frege133 40349 |
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