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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 6685), 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 6685 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
2 | eleq1a 2910 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ V → (V = (𝐹‘𝐴) → V ∈ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (V = (𝐹‘𝐴) → V ∈ V) |
4 | vprc 5221 | . . . 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 2831 | 1 ⊢ ((𝐹‘𝐴) = V → (𝐹‘𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-pr 4572 df-uni 4841 df-iota 6316 df-fv 6365 |
This theorem is referenced by: afvpcfv0 43352 |
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