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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 6881 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
2 | 1 | fveq1d 6891 | . 2 ⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = (∅‘𝐵)) |
3 | 0fv 6933 | . 2 ⊢ (∅‘𝐵) = ∅ | |
4 | 2, 3 | eqtrdi 2789 | 1 ⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4322 ‘cfv 6541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-dm 5686 df-iota 6493 df-fv 6549 |
This theorem is referenced by: elfv2ex 6935 itunitc1 10412 sralem 20783 sralemOLD 20784 srasca 20791 srascaOLD 20792 sravsca 20793 sravscaOLD 20794 sraip 20795 |
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