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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fveqvfvv | Structured version Visualization version GIF version |
Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 6915), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything (see pm2.21i 119). (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
fveqvfvv | ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6915 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
2 | eleq1a 2824 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ V → (V = (𝐹‘𝐴) → V ∈ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (V = (𝐹‘𝐴) → V ∈ V) |
4 | vprc 5319 | . . . 4 ⊢ ¬ V ∈ V | |
5 | 4 | pm2.21i 119 | . . 3 ⊢ (V ∈ V → (𝐹‘𝐴) = 𝐵) |
6 | 3, 5 | syl 17 | . 2 ⊢ (V = (𝐹‘𝐴) → (𝐹‘𝐴) = 𝐵) |
7 | 6 | eqcoms 2736 | 1 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-sn 4633 df-pr 4635 df-uni 4913 df-iota 6505 df-fv 6561 |
This theorem is referenced by: afvpcfv0 46573 |
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