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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 6898 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
| 2 | 1 | fveq1d 6908 | . 2 ⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = (∅‘𝐵)) | 
| 3 | 0fv 6950 | . 2 ⊢ (∅‘𝐵) = ∅ | |
| 4 | 2, 3 | eqtrdi 2793 | 1 ⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: elfv2ex 6952 itunitc1 10460 sralem 21175 sralemOLD 21176 srasca 21183 srascaOLD 21184 sravsca 21185 sravscaOLD 21186 sraip 21187 | 
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