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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 | vex 3331 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | elimasng 5637 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 〈𝐴, 𝑥〉 ∈ 𝐹)) | |
3 | df-br 4793 | . . . . . 6 ⊢ (𝐴𝐹𝑥 ↔ 〈𝐴, 𝑥〉 ∈ 𝐹) | |
4 | 2, 3 | syl6bbr 278 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
5 | 1, 4 | mpan2 709 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
6 | 5 | iotabidv 6021 | . . 3 ⊢ (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥)) |
7 | df-fv 6045 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
8 | 6, 7 | syl6reqr 2801 | . 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
9 | fvprc 6334 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
10 | snprc 4385 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | 10 | biimpi 206 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
12 | 11 | imaeq2d 5612 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅)) |
13 | ima0 5627 | . . . . . . 7 ⊢ (𝐹 “ ∅) = ∅ | |
14 | 12, 13 | syl6eq 2798 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅) |
15 | 14 | eleq2d 2813 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅)) |
16 | 15 | iotabidv 6021 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅)) |
17 | noel 4050 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
18 | 17 | nex 1868 | . . . . . 6 ⊢ ¬ ∃𝑥 𝑥 ∈ ∅ |
19 | euex 2619 | . . . . . 6 ⊢ (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅) | |
20 | 18, 19 | mto 188 | . . . . 5 ⊢ ¬ ∃!𝑥 𝑥 ∈ ∅ |
21 | iotanul 6015 | . . . . 5 ⊢ (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅) | |
22 | 20, 21 | ax-mp 5 | . . . 4 ⊢ (℩𝑥𝑥 ∈ ∅) = ∅ |
23 | 16, 22 | syl6eq 2798 | . . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅) |
24 | 9, 23 | eqtr4d 2785 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
25 | 8, 24 | pm2.61i 176 | 1 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 383 = wceq 1620 ∃wex 1841 ∈ wcel 2127 ∃!weu 2595 Vcvv 3328 ∅c0 4046 {csn 4309 〈cop 4315 class class class wbr 4792 “ cima 5257 ℩cio 5998 ‘cfv 6037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-br 4793 df-opab 4853 df-xp 5260 df-cnv 5262 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-iota 6000 df-fv 6045 |
This theorem is referenced by: dffv4 6337 fvco2 6423 shftval 13984 dffv5 32308 |
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