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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 5695 | . . . . 5 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 2, 3 | eqtr4i 2764 | . . . 4 ⊢ 𝐹 = (𝐴 × {𝐵}) |
5 | 4 | rneqi 5893 | . . 3 ⊢ ran 𝐹 = ran (𝐴 × {𝐵}) |
6 | rnxp 6123 | . . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵}) | |
7 | 5, 6 | eqtrid 2785 | . 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 1542 ≠ wne 2940 ∅c0 4283 {csn 4587 ↦ cmpt 5189 × cxp 5632 ran crn 5635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-mpt 5190 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 |
This theorem is referenced by: mptiffisupp 31654 qsalrel 40710 limsup0 44021 limsuppnfdlem 44028 limsup10ex 44100 liminf10ex 44101 fourierdlem60 44493 fourierdlem61 44494 sge0z 44702 |
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