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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 | df-fv 6571 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
2 | elimasng 6109 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 〈𝐴, 𝑥〉 ∈ 𝐹)) | |
3 | df-br 5149 | . . . . . 6 ⊢ (𝐴𝐹𝑥 ↔ 〈𝐴, 𝑥〉 ∈ 𝐹) | |
4 | 2, 3 | bitr4di 289 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
5 | 4 | elvd 3484 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
6 | 5 | iotabidv 6547 | . . 3 ⊢ (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥)) |
7 | 1, 6 | eqtr4id 2794 | . 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
8 | fvprc 6899 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
9 | snprc 4722 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | 9 | biimpi 216 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
11 | 10 | imaeq2d 6080 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅)) |
12 | ima0 6097 | . . . . . . 7 ⊢ (𝐹 “ ∅) = ∅ | |
13 | 11, 12 | eqtrdi 2791 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅) |
14 | 13 | eleq2d 2825 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅)) |
15 | 14 | iotabidv 6547 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅)) |
16 | noel 4344 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
17 | 16 | nex 1797 | . . . . . 6 ⊢ ¬ ∃𝑥 𝑥 ∈ ∅ |
18 | euex 2575 | . . . . . 6 ⊢ (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅) | |
19 | 17, 18 | mto 197 | . . . . 5 ⊢ ¬ ∃!𝑥 𝑥 ∈ ∅ |
20 | iotanul 6541 | . . . . 5 ⊢ (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅) | |
21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ (℩𝑥𝑥 ∈ ∅) = ∅ |
22 | 15, 21 | eqtrdi 2791 | . . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅) |
23 | 8, 22 | eqtr4d 2778 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
24 | 7, 23 | pm2.61i 182 | 1 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∃!weu 2566 Vcvv 3478 ∅c0 4339 {csn 4631 〈cop 4637 class class class wbr 5148 “ cima 5692 ℩cio 6514 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fv 6571 |
This theorem is referenced by: dffv4 6904 fvco2 7006 shftval 15110 dffv5 35906 |
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