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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 5578 | . . . . 5 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 2, 3 | eqtr4i 2824 | . . . 4 ⊢ 𝐹 = (𝐴 × {𝐵}) |
5 | 4 | rneqi 5771 | . . 3 ⊢ ran 𝐹 = ran (𝐴 × {𝐵}) |
6 | rnxp 5994 | . . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵}) | |
7 | 5, 6 | syl5eq 2845 | . 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 1538 ≠ wne 2987 ∅c0 4243 {csn 4525 ↦ cmpt 5110 × cxp 5517 ran crn 5520 |
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-v 3443 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-mpt 5111 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: qsalrel 39420 limsup0 42336 limsuppnfdlem 42343 limsup10ex 42415 liminf10ex 42416 fourierdlem60 42808 fourierdlem61 42809 sge0z 43014 |
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