Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvpcfv0 | Structured version Visualization version GIF version |
Description: If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afvpcfv0 | ⊢ ((𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfafv2 39861 | . . 3 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | 1 | eqeq1i 2615 | . 2 ⊢ ((𝐹'''𝐴) = V ↔ if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V) |
3 | eqcom 2617 | . . . 4 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V ↔ V = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)) | |
4 | eqif 4076 | . . . 4 ⊢ (V = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V))) | |
5 | 3, 4 | bitri 263 | . . 3 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V))) |
6 | fveqvfvv 39853 | . . . . . 6 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = ∅) | |
7 | 6 | eqcoms 2618 | . . . . 5 ⊢ (V = (𝐹‘𝐴) → (𝐹‘𝐴) = ∅) |
8 | 7 | adantl 481 | . . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) → (𝐹‘𝐴) = ∅) |
9 | fvfundmfvn0 6136 | . . . . . . 7 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
10 | df-dfat 39845 | . . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
11 | 9, 10 | sylibr 223 | . . . . . 6 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐹 defAt 𝐴) |
12 | 11 | necon1bi 2810 | . . . . 5 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹‘𝐴) = ∅) |
13 | 12 | adantr 480 | . . . 4 ⊢ ((¬ 𝐹 defAt 𝐴 ∧ V = V) → (𝐹‘𝐴) = ∅) |
14 | 8, 13 | jaoi 393 | . . 3 ⊢ (((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)) → (𝐹‘𝐴) = ∅) |
15 | 5, 14 | sylbi 206 | . 2 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V → (𝐹‘𝐴) = ∅) |
16 | 2, 15 | sylbi 206 | 1 ⊢ ((𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 ifcif 4036 {csn 4125 dom cdm 5038 ↾ cres 5040 Fun wfun 5798 ‘cfv 5804 defAt wdfat 39842 '''cafv 39843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 df-dfat 39845 df-afv 39846 |
This theorem is referenced by: afvfv0bi 39881 aovpcov0 39919 |
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