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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 6743. (Contributed by AV, 7-Sep-2022.) |
Ref | Expression |
---|---|
dfatsnafv2 | ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2828 | . . . 4 ⊢ (𝑦 = (𝐹''''𝐴) ↔ (𝐹''''𝐴) = 𝑦) | |
2 | dfatbrafv2b 43464 | . . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑦 ∈ V) → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) | |
3 | 2 | elvd 3500 | . . . 4 ⊢ (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
4 | 1, 3 | syl5bb 285 | . . 3 ⊢ (𝐹 defAt 𝐴 → (𝑦 = (𝐹''''𝐴) ↔ 𝐴𝐹𝑦)) |
5 | 4 | abbidv 2885 | . 2 ⊢ (𝐹 defAt 𝐴 → {𝑦 ∣ 𝑦 = (𝐹''''𝐴)} = {𝑦 ∣ 𝐴𝐹𝑦}) |
6 | df-sn 4568 | . . 3 ⊢ {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)} | |
7 | 6 | a1i 11 | . 2 ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)}) |
8 | dfdfat2 43347 | . . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) | |
9 | imasng 5951 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) | |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) |
11 | 8, 10 | sylbi 219 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) |
12 | 5, 7, 11 | 3eqtr4d 2866 | 1 ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃!weu 2653 {cab 2799 Vcvv 3494 {csn 4567 class class class wbr 5066 dom cdm 5555 “ cima 5558 defAt wdfat 43335 ''''cafv2 43427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-dfat 43338 df-afv2 43428 |
This theorem is referenced by: afv2co2 43476 |
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