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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnimasnd | Structured version Visualization version GIF version |
Description: The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.) |
Ref | Expression |
---|---|
fnimasnd.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
fnimasnd.2 | ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
Ref | Expression |
---|---|
fnimasnd | ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnimasnd.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | fnimasnd.2 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐴) | |
3 | fneu 6338 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴) → ∃!𝑥 𝑆𝐹𝑥) | |
4 | 1, 2, 3 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∃!𝑥 𝑆𝐹𝑥) |
5 | sniota 6223 | . . 3 ⊢ (∃!𝑥 𝑆𝐹𝑥 → {𝑥 ∣ 𝑆𝐹𝑥} = {(℩𝑥𝑆𝐹𝑥)}) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → {𝑥 ∣ 𝑆𝐹𝑥} = {(℩𝑥𝑆𝐹𝑥)}) |
7 | fnrel 6331 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
8 | 1, 7 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝐹) |
9 | relimasn 5835 | . . 3 ⊢ (Rel 𝐹 → (𝐹 “ {𝑆}) = {𝑥 ∣ 𝑆𝐹𝑥}) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → (𝐹 “ {𝑆}) = {𝑥 ∣ 𝑆𝐹𝑥}) |
11 | df-fv 6240 | . . . 4 ⊢ (𝐹‘𝑆) = (℩𝑥𝑆𝐹𝑥) | |
12 | 11 | sneqi 4489 | . . 3 ⊢ {(𝐹‘𝑆)} = {(℩𝑥𝑆𝐹𝑥)} |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → {(𝐹‘𝑆)} = {(℩𝑥𝑆𝐹𝑥)}) |
14 | 6, 10, 13 | 3eqtr4d 2843 | 1 ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 ∃!weu 2613 {cab 2777 {csn 4478 class class class wbr 4968 “ cima 5453 Rel wrel 5455 ℩cio 6194 Fn wfn 6227 ‘cfv 6232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-fv 6240 |
This theorem is referenced by: (None) |
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