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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 6569 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
| 2 | elimasng 6107 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 〈𝐴, 𝑥〉 ∈ 𝐹)) | |
| 3 | df-br 5144 | . . . . . 6 ⊢ (𝐴𝐹𝑥 ↔ 〈𝐴, 𝑥〉 ∈ 𝐹) | |
| 4 | 2, 3 | bitr4di 289 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
| 5 | 4 | elvd 3486 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
| 6 | 5 | iotabidv 6545 | . . 3 ⊢ (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥)) |
| 7 | 1, 6 | eqtr4id 2796 | . 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
| 8 | fvprc 6898 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
| 9 | snprc 4717 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 10 | 9 | biimpi 216 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
| 11 | 10 | imaeq2d 6078 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅)) |
| 12 | ima0 6095 | . . . . . . 7 ⊢ (𝐹 “ ∅) = ∅ | |
| 13 | 11, 12 | eqtrdi 2793 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅) |
| 14 | 13 | eleq2d 2827 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅)) |
| 15 | 14 | iotabidv 6545 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅)) |
| 16 | noel 4338 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
| 17 | 16 | nex 1800 | . . . . . 6 ⊢ ¬ ∃𝑥 𝑥 ∈ ∅ |
| 18 | euex 2577 | . . . . . 6 ⊢ (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅) | |
| 19 | 17, 18 | mto 197 | . . . . 5 ⊢ ¬ ∃!𝑥 𝑥 ∈ ∅ |
| 20 | iotanul 6539 | . . . . 5 ⊢ (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅) | |
| 21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ (℩𝑥𝑥 ∈ ∅) = ∅ |
| 22 | 15, 21 | eqtrdi 2793 | . . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅) |
| 23 | 8, 22 | eqtr4d 2780 | . 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 1540 ∃wex 1779 ∈ wcel 2108 ∃!weu 2568 Vcvv 3480 ∅c0 4333 {csn 4626 〈cop 4632 class class class wbr 5143 “ cima 5688 ℩cio 6512 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: dffv4 6903 fvco2 7006 shftval 15113 dffv5 35925 |
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