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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 6454 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴) → ∃!𝑥 𝑆𝐹𝑥) | |
4 | 1, 2, 3 | syl2anc 586 | . . 3 ⊢ (𝜑 → ∃!𝑥 𝑆𝐹𝑥) |
5 | sniota 6339 | . . 3 ⊢ (∃!𝑥 𝑆𝐹𝑥 → {𝑥 ∣ 𝑆𝐹𝑥} = {(℩𝑥𝑆𝐹𝑥)}) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → {𝑥 ∣ 𝑆𝐹𝑥} = {(℩𝑥𝑆𝐹𝑥)}) |
7 | fnrel 6447 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
8 | 1, 7 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝐹) |
9 | relimasn 5945 | . . 3 ⊢ (Rel 𝐹 → (𝐹 “ {𝑆}) = {𝑥 ∣ 𝑆𝐹𝑥}) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → (𝐹 “ {𝑆}) = {𝑥 ∣ 𝑆𝐹𝑥}) |
11 | df-fv 6356 | . . . 4 ⊢ (𝐹‘𝑆) = (℩𝑥𝑆𝐹𝑥) | |
12 | 11 | sneqi 4571 | . . 3 ⊢ {(𝐹‘𝑆)} = {(℩𝑥𝑆𝐹𝑥)} |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → {(𝐹‘𝑆)} = {(℩𝑥𝑆𝐹𝑥)}) |
14 | 6, 10, 13 | 3eqtr4d 2865 | 1 ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∃!weu 2652 {cab 2798 {csn 4560 class class class wbr 5059 “ cima 5551 Rel wrel 5553 ℩cio 6305 Fn wfn 6343 ‘cfv 6348 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: (None) |
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