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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 3504 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | dfima2 5917 | . . 3 ⊢ (𝑅 “ {𝐴}) = {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} | |
3 | breq1 5055 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦)) | |
4 | 3 | rexsng 4600 | . . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦)) |
5 | 4 | abbidv 2885 | . . 3 ⊢ (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ {𝐴}𝑥𝑅𝑦} = {𝑦 ∣ 𝐴𝑅𝑦}) |
6 | 2, 5 | syl5eq 2868 | . 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 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 Vcvv 3486 {csn 4553 class class class wbr 5052 “ cima 5544 |
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 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-xp 5547 df-cnv 5549 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 |
This theorem is referenced by: relimasn 5938 elimasn 5940 args 5943 suppvalbr 7820 dfec2 8278 dfac3 9533 shftfib 14416 areacirclem5 35018 dfcoll2 40678 dfatsnafv2 43541 |
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