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Mirrors > Home > MPE Home > Th. List > imasng | Structured version Visualization version GIF version |
Description: The image of a singleton. (Contributed by NM, 8-May-2005.) |
Ref | Expression |
---|---|
imasng | ⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3364 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | dfima2 5609 | . . 3 ⊢ (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} | |
3 | breq1 4789 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦)) | |
4 | 3 | rexsng 4357 | . . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦)) |
5 | 4 | abbidv 2890 | . . 3 ⊢ (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦 ∣ 𝐴𝑅𝑦}) |
6 | 2, 5 | syl5eq 2817 | . 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
7 | 1, 6 | syl 17 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 {cab 2757 ∃wrex 3062 Vcvv 3351 {csn 4316 class class class wbr 4786 “ cima 5252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-xp 5255 df-cnv 5257 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 |
This theorem is referenced by: relimasn 5629 elimasn 5631 args 5634 suppvalbr 7450 dfec2 7899 dfac3 9144 shftfib 14020 areacirclem5 33836 |
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