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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 6898), 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 6898 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
2 | eleq1a 2822 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ V → (V = (𝐹‘𝐴) → V ∈ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (V = (𝐹‘𝐴) → V ∈ V) |
4 | vprc 5308 | . . . 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 2734 | 1 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ‘cfv 6537 |
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 2697 ax-sep 5292 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-sn 4624 df-pr 4626 df-uni 4903 df-iota 6489 df-fv 6545 |
This theorem is referenced by: afvpcfv0 46426 |
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