| 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 4643 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
| 2 | 1 | fveq2d 6910 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴)) |
| 3 | 2 | mpteq2ia 5245 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)) |
| 4 | rnsnf.2 | . . . . 5 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) | |
| 5 | 4 | feqmptd 6977 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥))) |
| 6 | rnsnf.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 7 | fvexd 6921 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ V) | |
| 8 | fmptsn 7187 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ V) → {〈𝐴, (𝐹‘𝐴)〉} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))) | |
| 9 | 6, 7, 8 | syl2anc 584 | . . . 4 ⊢ (𝜑 → {〈𝐴, (𝐹‘𝐴)〉} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))) |
| 10 | 3, 5, 9 | 3eqtr4a 2803 | . . 3 ⊢ (𝜑 → 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
| 11 | 10 | rneqd 5949 | . 2 ⊢ (𝜑 → ran 𝐹 = ran {〈𝐴, (𝐹‘𝐴)〉}) |
| 12 | rnsnopg 6241 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, (𝐹‘𝐴)〉} = {(𝐹‘𝐴)}) | |
| 13 | 6, 12 | syl 17 | . 2 ⊢ (𝜑 → ran {〈𝐴, (𝐹‘𝐴)〉} = {(𝐹‘𝐴)}) |
| 14 | 11, 13 | eqtrd 2777 | 1 ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ↦ cmpt 5225 ran crn 5686 ⟶wf 6557 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 |
| This theorem is referenced by: fsneqrn 45216 unirnmapsn 45219 sge0sn 46394 |
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