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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 4722 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | imaeq2 6076 | . . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅)) | |
3 | 1, 2 | sylbi 217 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅)) |
4 | ima0 6097 | . . . . . 6 ⊢ (𝑅 “ ∅) = ∅ | |
5 | 3, 4 | eqtrdi 2791 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅) |
6 | 5 | adantl 481 | . . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅) |
7 | brrelex1 5742 | . . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑦) → 𝐴 ∈ V) | |
8 | 7 | stoic1a 1769 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦) |
9 | 8 | nexdv 1934 | . . . . 5 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦) |
10 | abn0 4391 | . . . . . 6 ⊢ ({𝑦 ∣ 𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦) | |
11 | 10 | necon1bbii 2988 | . . . . 5 ⊢ (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦 ∣ 𝐴𝑅𝑦} = ∅) |
12 | 9, 11 | sylib 218 | . . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦 ∣ 𝐴𝑅𝑦} = ∅) |
13 | 6, 12 | eqtr4d 2778 | . . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
14 | 13 | ex 412 | . 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})) |
15 | imasng 6104 | . 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 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 Vcvv 3478 ∅c0 4339 {csn 4631 class class class wbr 5148 “ cima 5692 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: elrelimasn 6106 predep 6353 funfv2 6997 mapsnd 8925 nznngen 44312 nzss 44313 hashnzfz 44316 |
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