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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatsnafv2 | Structured version Visualization version GIF version |
Description: Singleton of function value, analogous to fnsnfv 6964. (Contributed by AV, 7-Sep-2022.) |
Ref | Expression |
---|---|
dfatsnafv2 | ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2733 | . . . 4 ⊢ (𝑦 = (𝐹''''𝐴) ↔ (𝐹''''𝐴) = 𝑦) | |
2 | dfatbrafv2b 46530 | . . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑦 ∈ V) → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) | |
3 | 2 | elvd 3475 | . . . 4 ⊢ (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
4 | 1, 3 | bitrid 283 | . . 3 ⊢ (𝐹 defAt 𝐴 → (𝑦 = (𝐹''''𝐴) ↔ 𝐴𝐹𝑦)) |
5 | 4 | abbidv 2795 | . 2 ⊢ (𝐹 defAt 𝐴 → {𝑦 ∣ 𝑦 = (𝐹''''𝐴)} = {𝑦 ∣ 𝐴𝐹𝑦}) |
6 | df-sn 4624 | . . 3 ⊢ {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)} | |
7 | 6 | a1i 11 | . 2 ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)}) |
8 | dfdfat2 46413 | . . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) | |
9 | imasng 6076 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) |
11 | 8, 10 | sylbi 216 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) |
12 | 5, 7, 11 | 3eqtr4d 2776 | 1 ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃!weu 2556 {cab 2703 Vcvv 3468 {csn 4623 class class class wbr 5141 dom cdm 5669 “ cima 5672 defAt wdfat 46401 ''''cafv2 46493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-dfat 46404 df-afv2 46494 |
This theorem is referenced by: afv2co2 46542 |
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