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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 5614 | . . . . 5 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 2, 3 | eqtr4i 2847 | . . . 4 ⊢ 𝐹 = (𝐴 × {𝐵}) |
5 | 4 | rneqi 5807 | . . 3 ⊢ ran 𝐹 = ran (𝐴 × {𝐵}) |
6 | rnxp 6027 | . . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵}) | |
7 | 5, 6 | syl5eq 2868 | . 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 1537 ≠ wne 3016 ∅c0 4291 {csn 4567 ↦ cmpt 5146 × cxp 5553 ran crn 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: qsalrel 39145 limsup0 41995 limsuppnfdlem 42002 limsup10ex 42074 liminf10ex 42075 fourierdlem60 42471 fourierdlem61 42472 sge0z 42677 |
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