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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 5740 | . . . . 5 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 2, 3 | eqtr4i 2759 | . . . 4 ⊢ 𝐹 = (𝐴 × {𝐵}) |
5 | 4 | rneqi 5939 | . . 3 ⊢ ran 𝐹 = ran (𝐴 × {𝐵}) |
6 | rnxp 6174 | . . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵}) | |
7 | 5, 6 | eqtrid 2780 | . 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 1534 ≠ wne 2937 ∅c0 4323 {csn 4629 ↦ cmpt 5231 × cxp 5676 ran crn 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 |
This theorem is referenced by: mptiffisupp 32486 qsalrel 41731 limsup0 45082 limsuppnfdlem 45089 limsup10ex 45161 liminf10ex 45162 fourierdlem60 45554 fourierdlem61 45555 sge0z 45763 |
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