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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 4613 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | imaeq2 5892 | . . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 “ {𝐴}) = (𝑅 “ ∅)) | |
3 | 1, 2 | sylbi 220 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = (𝑅 “ ∅)) |
4 | ima0 5912 | . . . . . 6 ⊢ (𝑅 “ ∅) = ∅ | |
5 | 3, 4 | eqtrdi 2849 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = ∅) |
6 | 5 | adantl 485 | . . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = ∅) |
7 | brrelex1 5569 | . . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑦) → 𝐴 ∈ V) | |
8 | 7 | stoic1a 1774 | . . . . . 6 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝑦) |
9 | 8 | nexdv 1937 | . . . . 5 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ ∃𝑦 𝐴𝑅𝑦) |
10 | abn0 4290 | . . . . . 6 ⊢ ({𝑦 ∣ 𝐴𝑅𝑦} ≠ ∅ ↔ ∃𝑦 𝐴𝑅𝑦) | |
11 | 10 | necon1bbii 3036 | . . . . 5 ⊢ (¬ ∃𝑦 𝐴𝑅𝑦 ↔ {𝑦 ∣ 𝐴𝑅𝑦} = ∅) |
12 | 9, 11 | sylib 221 | . . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → {𝑦 ∣ 𝐴𝑅𝑦} = ∅) |
13 | 6, 12 | eqtr4d 2836 | . . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
14 | 13 | ex 416 | . 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})) |
15 | imasng 5918 | . 2 ⊢ (𝐴 ∈ V → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) | |
16 | 14, 15 | pm2.61d2 184 | 1 ⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 Vcvv 3441 ∅c0 4243 {csn 4525 class class class wbr 5030 “ cima 5522 Rel wrel 5524 |
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-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 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-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: elrelimasn 5920 predep 6142 fnsnfv 6718 funfv2 6726 mapsnd 8433 fnimasnd 39415 nznngen 41020 nzss 41021 hashnzfz 41024 |
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