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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptc | Structured version Visualization version GIF version |
Description: Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
rnmptc.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
rnmptc.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
rnmptc.a | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Ref | Expression |
---|---|
rnmptc | ⊢ (𝜑 → ran 𝐹 = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmptc.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | fconstmpt 5411 | . . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | eqtr4i 2805 | . 2 ⊢ 𝐹 = (𝐴 × {𝐵}) |
4 | rnmptc.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
5 | 4, 1 | fmptd 6648 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
6 | 5 | ffnd 6292 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
7 | rnmptc.a | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
8 | fconst5 6743 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵})) | |
9 | 6, 7, 8 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵})) |
10 | 3, 9 | mpbii 225 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∅c0 4141 {csn 4398 ↦ cmpt 4965 × cxp 5353 ran crn 5356 Fn wfn 6130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fo 6141 df-fv 6143 |
This theorem is referenced by: limsup0 40834 limsuppnfdlem 40841 limsup10ex 40913 liminf10ex 40914 fourierdlem60 41310 fourierdlem61 41311 sge0z 41516 |
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