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Mirrors > Home > MPE Home > Th. List > fv2prc | Structured version Visualization version GIF version |
Description: A function value of a function value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.) |
Ref | Expression |
---|---|
fv2prc | ⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvprc 6665 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
2 | 1 | fveq1d 6674 | . 2 ⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = (∅‘𝐵)) |
3 | 0fv 6711 | . 2 ⊢ (∅‘𝐵) = ∅ | |
4 | 2, 3 | syl6eq 2874 | 1 ⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 ‘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-nul 5212 ax-pow 5268 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 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-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-dm 5567 df-iota 6316 df-fv 6365 |
This theorem is referenced by: elfv2ex 6713 itunitc1 9844 sralem 19951 srasca 19955 sravsca 19956 sraip 19957 |
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