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| Mirrors > Home > MPE Home > Th. List > rnmptc | Structured version Visualization version GIF version | ||
| Description: Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Remove extra hypothesis. (Revised by SN, 17-Apr-2024.) |
| Ref | Expression |
|---|---|
| rnmptc.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| rnmptc.a | ⊢ (𝜑 → 𝐴 ≠ ∅) |
| Ref | Expression |
|---|---|
| rnmptc | ⊢ (𝜑 → ran 𝐹 = {𝐵}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rnmptc.a | . 2 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
| 2 | rnmptc.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 3 | fconstmpt 5747 | . . . . 5 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 4 | 2, 3 | eqtr4i 2768 | . . . 4 ⊢ 𝐹 = (𝐴 × {𝐵}) |
| 5 | 4 | rneqi 5948 | . . 3 ⊢ ran 𝐹 = ran (𝐴 × {𝐵}) |
| 6 | rnxp 6190 | . . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵}) | |
| 7 | 5, 6 | eqtrid 2789 | . 2 ⊢ (𝐴 ≠ ∅ → ran 𝐹 = {𝐵}) |
| 8 | 1, 7 | syl 17 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐵}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ≠ wne 2940 ∅c0 4333 {csn 4626 ↦ cmpt 5225 × cxp 5683 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: mptiffisupp 32702 qsalrel 42281 limsup0 45709 limsuppnfdlem 45716 limsup10ex 45788 liminf10ex 45789 fourierdlem60 46181 fourierdlem61 46182 sge0z 46390 |
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