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 43405 | . . 3 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
2 | 1 | eqeq1i 2825 | . 2 ⊢ ((𝐹'''𝐴) = V ↔ if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V) |
3 | eqcom 2827 | . . . 4 ⊢ (if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V ↔ V = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)) | |
4 | eqif 4500 | . . . 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 43349 | . . . . . 6 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = ∅) | |
7 | 6 | eqcoms 2828 | . . . . 5 ⊢ (V = (𝐹‘𝐴) → (𝐹‘𝐴) = ∅) |
8 | 7 | adantl 484 | . . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ V = (𝐹‘𝐴)) → (𝐹‘𝐴) = ∅) |
9 | fvfundmfvn0 6701 | . . . . . . 7 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
10 | df-dfat 43392 | . . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
11 | 9, 10 | sylibr 236 | . . . . . 6 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐹 defAt 𝐴) |
12 | 11 | necon1bi 3043 | . . . . 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 1536 ∈ wcel 2113 ≠ wne 3015 Vcvv 3491 ∅c0 4284 ifcif 4460 {csn 4560 dom cdm 5548 ↾ cres 5550 Fun wfun 6342 ‘cfv 6348 defAt wdfat 43389 '''cafv 43390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4870 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-aiota 43359 df-dfat 43392 df-afv 43393 |
This theorem is referenced by: afvfv0bi 43425 aovpcov0 43463 |
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