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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 5034 | . . 3 ⊢ (𝑥 = 𝐶 → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶)) | |
3 | elimasn.1 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | imasng 5918 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥}) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥} |
6 | 1, 2, 5 | elab2 3618 | . 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶) |
7 | df-br 5031 | . 2 ⊢ (𝐵𝐴𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐴) | |
8 | 6, 7 | bitri 278 | 1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 〈𝐵, 𝐶〉 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2776 Vcvv 3441 {csn 4525 〈cop 4531 class class class wbr 5030 “ cima 5522 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: elimasng 5922 dfco2 6065 dfco2a 6066 ressn 6104 funfvima3 6976 frxp 7803 marypha1lem 8881 gsum2dlem1 19083 gsum2dlem2 19084 gsum2d 19085 gsum2d2 19087 ovoliunlem1 24106 iunsnima 30382 dfcnv2 30439 gsummpt2co 30733 gsummpt2d 30734 dmscut 33385 scutf 33386 funpartfun 33517 areaquad 40166 dffrege76 40640 frege97 40661 frege98 40662 frege109 40673 frege110 40674 frege131 40695 frege133 40697 |
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