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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 43688 | . . 3 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | 1 | eqeq1i 2803 | . 2 ⊢ ((𝐹'''𝐴) = V ↔ if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V) |
3 | eqcom 2805 | . . . 4 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V ↔ V = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)) | |
4 | eqif 4465 | . . . 4 ⊢ (V = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V))) | |
5 | 3, 4 | bitri 278 | . . 3 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V))) |
6 | fveqvfvv 43632 | . . . . . 6 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = ∅) | |
7 | 6 | eqcoms 2806 | . . . . 5 ⊢ (V = (𝐹‘𝐴) → (𝐹‘𝐴) = ∅) |
8 | 7 | adantl 485 | . . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) → (𝐹‘𝐴) = ∅) |
9 | fvfundmfvn0 6683 | . . . . . . 7 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
10 | df-dfat 43675 | . . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
11 | 9, 10 | sylibr 237 | . . . . . 6 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐹 defAt 𝐴) |
12 | 11 | necon1bi 3015 | . . . . 5 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹‘𝐴) = ∅) |
13 | 12 | adantr 484 | . . . 4 ⊢ ((¬ 𝐹 defAt 𝐴 ∧ V = V) → (𝐹‘𝐴) = ∅) |
14 | 8, 13 | jaoi 854 | . . 3 ⊢ (((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)) → (𝐹‘𝐴) = ∅) |
15 | 5, 14 | sylbi 220 | . 2 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V → (𝐹‘𝐴) = ∅) |
16 | 2, 15 | sylbi 220 | 1 ⊢ ((𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 ifcif 4425 {csn 4525 dom cdm 5519 ↾ cres 5521 Fun wfun 6318 ‘cfv 6324 defAt wdfat 43672 '''cafv 43673 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-aiota 43642 df-dfat 43675 df-afv 43676 |
This theorem is referenced by: afvfv0bi 43708 aovpcov0 43746 |
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