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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 45517 | . . 3 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | 1 | eqeq1i 2736 | . 2 ⊢ ((𝐹'''𝐴) = V ↔ if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V) |
3 | eqcom 2738 | . . . 4 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V ↔ V = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)) | |
4 | eqif 4547 | . . . 4 ⊢ (V = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V))) | |
5 | 3, 4 | bitri 274 | . . 3 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V))) |
6 | fveqvfvv 45427 | . . . . . 6 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = ∅) | |
7 | 6 | eqcoms 2739 | . . . . 5 ⊢ (V = (𝐹‘𝐴) → (𝐹‘𝐴) = ∅) |
8 | 7 | adantl 482 | . . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) → (𝐹‘𝐴) = ∅) |
9 | fvfundmfvn0 6905 | . . . . . . 7 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
10 | df-dfat 45504 | . . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
11 | 9, 10 | sylibr 233 | . . . . . 6 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐹 defAt 𝐴) |
12 | 11 | necon1bi 2968 | . . . . 5 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹‘𝐴) = ∅) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((¬ 𝐹 defAt 𝐴 ∧ V = V) → (𝐹‘𝐴) = ∅) |
14 | 8, 13 | jaoi 855 | . . 3 ⊢ (((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)) → (𝐹‘𝐴) = ∅) |
15 | 5, 14 | sylbi 216 | . 2 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V → (𝐹‘𝐴) = ∅) |
16 | 2, 15 | sylbi 216 | 1 ⊢ ((𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 Vcvv 3459 ∅c0 4302 ifcif 4506 {csn 4606 dom cdm 5653 ↾ cres 5655 Fun wfun 6510 ‘cfv 6516 defAt wdfat 45501 '''cafv 45502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pr 5404 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-br 5126 df-opab 5188 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-res 5665 df-iota 6468 df-fun 6518 df-fv 6524 df-aiota 45470 df-dfat 45504 df-afv 45505 |
This theorem is referenced by: afvfv0bi 45537 aovpcov0 45575 |
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