![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dffv3 | Structured version Visualization version GIF version |
Description: A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.) |
Ref | Expression |
---|---|
dffv3 | ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 6332 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
2 | elimasng 5922 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 〈𝐴, 𝑥〉 ∈ 𝐹)) | |
3 | df-br 5031 | . . . . . 6 ⊢ (𝐴𝐹𝑥 ↔ 〈𝐴, 𝑥〉 ∈ 𝐹) | |
4 | 2, 3 | syl6bbr 292 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
5 | 4 | elvd 3447 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
6 | 5 | iotabidv 6308 | . . 3 ⊢ (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥)) |
7 | 1, 6 | eqtr4id 2852 | . 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
8 | fvprc 6638 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
9 | snprc 4613 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | 9 | biimpi 219 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
11 | 10 | imaeq2d 5896 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅)) |
12 | ima0 5912 | . . . . . . 7 ⊢ (𝐹 “ ∅) = ∅ | |
13 | 11, 12 | eqtrdi 2849 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅) |
14 | 13 | eleq2d 2875 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅)) |
15 | 14 | iotabidv 6308 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅)) |
16 | noel 4247 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
17 | 16 | nex 1802 | . . . . . 6 ⊢ ¬ ∃𝑥 𝑥 ∈ ∅ |
18 | euex 2637 | . . . . . 6 ⊢ (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅) | |
19 | 17, 18 | mto 200 | . . . . 5 ⊢ ¬ ∃!𝑥 𝑥 ∈ ∅ |
20 | iotanul 6302 | . . . . 5 ⊢ (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅) | |
21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ (℩𝑥𝑥 ∈ ∅) = ∅ |
22 | 15, 21 | eqtrdi 2849 | . . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅) |
23 | 8, 22 | eqtr4d 2836 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
24 | 7, 23 | pm2.61i 185 | 1 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃!weu 2628 Vcvv 3441 ∅c0 4243 {csn 4525 〈cop 4531 class class class wbr 5030 “ cima 5522 ℩cio 6281 ‘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-pow 5231 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-ral 3111 df-rex 3112 df-rab 3115 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-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fv 6332 |
This theorem is referenced by: dffv4 6642 fvco2 6735 shftval 14425 dffv5 33498 |
Copyright terms: Public domain | W3C validator |