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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnsnf | Structured version Visualization version GIF version |
Description: The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
rnsnf.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
rnsnf.2 | ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) |
Ref | Expression |
---|---|
rnsnf | ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 4227 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
2 | 1 | fveq2d 6233 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴)) |
3 | 2 | mpteq2ia 4773 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)) |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))) |
5 | rnsnf.2 | . . . . 5 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) | |
6 | 5 | feqmptd 6288 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥))) |
7 | rnsnf.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | fvexd 6241 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ V) | |
9 | fmptsn 6474 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ V) → {〈𝐴, (𝐹‘𝐴)〉} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))) | |
10 | 7, 8, 9 | syl2anc 694 | . . . 4 ⊢ (𝜑 → {〈𝐴, (𝐹‘𝐴)〉} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))) |
11 | 4, 6, 10 | 3eqtr4d 2695 | . . 3 ⊢ (𝜑 → 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
12 | 11 | rneqd 5385 | . 2 ⊢ (𝜑 → ran 𝐹 = ran {〈𝐴, (𝐹‘𝐴)〉}) |
13 | rnsnopg 5650 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, (𝐹‘𝐴)〉} = {(𝐹‘𝐴)}) | |
14 | 7, 13 | syl 17 | . 2 ⊢ (𝜑 → ran {〈𝐴, (𝐹‘𝐴)〉} = {(𝐹‘𝐴)}) |
15 | 12, 14 | eqtrd 2685 | 1 ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 {csn 4210 〈cop 4216 ↦ cmpt 4762 ran crn 5144 ⟶wf 5922 ‘cfv 5926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 |
This theorem is referenced by: fsneqrn 39717 unirnmapsn 39720 sge0sn 40914 |
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