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Mirrors > Home > MPE Home > Th. List > relimasn | Structured version Visualization version GIF version |
Description: The image of a singleton. (Contributed by NM, 20-May-1998.) |
Ref | Expression |
---|---|
relimasn | ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snprc 4726 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | imaeq2 6064 | . . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅)) | |
3 | 1, 2 | sylbi 216 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅)) |
4 | ima0 6085 | . . . . . 6 ⊢ (𝑅 “ ∅) = ∅ | |
5 | 3, 4 | eqtrdi 2784 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅) |
6 | 5 | adantl 480 | . . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅) |
7 | brrelex1 5735 | . . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑦) → 𝐴 ∈ V) | |
8 | 7 | stoic1a 1766 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦) |
9 | 8 | nexdv 1931 | . . . . 5 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦) |
10 | abn0 4384 | . . . . . 6 ⊢ ({𝑦 ∣ 𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦) | |
11 | 10 | necon1bbii 2987 | . . . . 5 ⊢ (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦 ∣ 𝐴𝑅𝑦} = ∅) |
12 | 9, 11 | sylib 217 | . . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦 ∣ 𝐴𝑅𝑦} = ∅) |
13 | 6, 12 | eqtr4d 2771 | . . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
14 | 13 | ex 411 | . 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})) |
15 | imasng 6092 | . 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) | |
16 | 14, 15 | pm2.61d2 181 | 1 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2705 Vcvv 3473 ∅c0 4326 {csn 4632 class class class wbr 5152 “ cima 5685 Rel wrel 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 |
This theorem is referenced by: elrelimasn 6094 predep 6341 fnsnfvOLD 6983 funfv2 6991 mapsnd 8913 nznngen 43802 nzss 43803 hashnzfz 43806 |
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