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 43325 | . . 3 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | 1 | eqeq1i 2826 | . 2 ⊢ ((𝐹'''𝐴) = V ↔ if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V) |
3 | eqcom 2828 | . . . 4 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V ↔ V = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)) | |
4 | eqif 4506 | . . . 4 ⊢ (V = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V))) | |
5 | 3, 4 | bitri 277 | . . 3 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V))) |
6 | fveqvfvv 43269 | . . . . . 6 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = ∅) | |
7 | 6 | eqcoms 2829 | . . . . 5 ⊢ (V = (𝐹‘𝐴) → (𝐹‘𝐴) = ∅) |
8 | 7 | adantl 484 | . . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) → (𝐹‘𝐴) = ∅) |
9 | fvfundmfvn0 6702 | . . . . . . 7 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
10 | df-dfat 43312 | . . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
11 | 9, 10 | sylibr 236 | . . . . . 6 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐹 defAt 𝐴) |
12 | 11 | necon1bi 3044 | . . . . 5 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹‘𝐴) = ∅) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((¬ 𝐹 defAt 𝐴 ∧ V = V) → (𝐹‘𝐴) = ∅) |
14 | 8, 13 | jaoi 853 | . . 3 ⊢ (((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)) → (𝐹‘𝐴) = ∅) |
15 | 5, 14 | sylbi 219 | . 2 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V → (𝐹‘𝐴) = ∅) |
16 | 2, 15 | sylbi 219 | 1 ⊢ ((𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∅c0 4290 ifcif 4466 {csn 4560 dom cdm 5549 ↾ cres 5551 Fun wfun 6343 ‘cfv 6349 defAt wdfat 43309 '''cafv 43310 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 df-aiota 43279 df-dfat 43312 df-afv 43313 |
This theorem is referenced by: afvfv0bi 43345 aovpcov0 43383 |
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