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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 6113), 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 115). (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
fveqvfvv | ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6113 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
2 | eleq1a 2683 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ V → (V = (𝐹‘𝐴) → V ∈ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (V = (𝐹‘𝐴) → V ∈ V) |
4 | vprc 4724 | . . . 4 ⊢ ¬ V ∈ V | |
5 | 4 | pm2.21i 115 | . . 3 ⊢ (V ∈ V → (𝐹‘𝐴) = 𝐵) |
6 | 3, 5 | syl 17 | . 2 ⊢ (V = (𝐹‘𝐴) → (𝐹‘𝐴) = 𝐵) |
7 | 6 | eqcoms 2618 | 1 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-fv 5812 |
This theorem is referenced by: afvpcfv0 39875 |
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