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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 4542 | . . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
2 | 1 | fveq2d 6649 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴)) |
3 | 2 | mpteq2ia 5121 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴)) |
4 | rnsnf.2 | . . . . 5 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) | |
5 | 4 | feqmptd 6708 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝑥))) |
6 | rnsnf.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | fvexd 6660 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ V) | |
8 | fmptsn 6906 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ∈ V) → {〈𝐴, (𝐹‘𝐴)〉} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))) | |
9 | 6, 7, 8 | syl2anc 587 | . . . 4 ⊢ (𝜑 → {〈𝐴, (𝐹‘𝐴)〉} = (𝑥 ∈ {𝐴} ↦ (𝐹‘𝐴))) |
10 | 3, 5, 9 | 3eqtr4a 2859 | . . 3 ⊢ (𝜑 → 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
11 | 10 | rneqd 5772 | . 2 ⊢ (𝜑 → ran 𝐹 = ran {〈𝐴, (𝐹‘𝐴)〉}) |
12 | rnsnopg 6045 | . . 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 2833 | 1 ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 〈cop 4531 ↦ cmpt 5110 ran crn 5520 ⟶wf 6320 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 |
This theorem is referenced by: fsneqrn 41840 unirnmapsn 41843 sge0sn 43018 |
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