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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 6552 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
2 | elimasng 6088 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)) | |
3 | df-br 5150 | . . . . . 6 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹) | |
4 | 2, 3 | bitr4di 289 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
5 | 4 | elvd 3482 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
6 | 5 | iotabidv 6528 | . . 3 ⊢ (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥)) |
7 | 1, 6 | eqtr4id 2792 | . 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
8 | fvprc 6884 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
9 | snprc 4722 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | 9 | biimpi 215 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
11 | 10 | imaeq2d 6060 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅)) |
12 | ima0 6077 | . . . . . . 7 ⊢ (𝐹 “ ∅) = ∅ | |
13 | 11, 12 | eqtrdi 2789 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅) |
14 | 13 | eleq2d 2820 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅)) |
15 | 14 | iotabidv 6528 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅)) |
16 | noel 4331 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
17 | 16 | nex 1803 | . . . . . 6 ⊢ ¬ ∃𝑥 𝑥 ∈ ∅ |
18 | euex 2572 | . . . . . 6 ⊢ (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅) | |
19 | 17, 18 | mto 196 | . . . . 5 ⊢ ¬ ∃!𝑥 𝑥 ∈ ∅ |
20 | iotanul 6522 | . . . . 5 ⊢ (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅) | |
21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ (℩𝑥𝑥 ∈ ∅) = ∅ |
22 | 15, 21 | eqtrdi 2789 | . . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅) |
23 | 8, 22 | eqtr4d 2776 | . 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 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃!weu 2563 Vcvv 3475 ∅c0 4323 {csn 4629 ⟨cop 4635 class class class wbr 5149 “ cima 5680 ℩cio 6494 ‘cfv 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fv 6552 |
This theorem is referenced by: dffv4 6889 fvco2 6989 shftval 15021 dffv5 34896 |
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