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Mirrors > Home > MPE Home > Th. List > rnfvprc | Structured version Visualization version GIF version |
Description: The range of a function value at a proper class is empty. (Contributed by AV, 20-Aug-2022.) |
Ref | Expression |
---|---|
rnfvprc.y | ⊢ 𝑌 = (𝐹‘𝑋) |
Ref | Expression |
---|---|
rnfvprc | ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnfvprc.y | . . . 4 ⊢ 𝑌 = (𝐹‘𝑋) | |
2 | fvprc 6881 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
3 | 1, 2 | eqtrid 2785 | . . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅) |
4 | 3 | rneqd 5936 | . 2 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ran ∅) |
5 | rn0 5924 | . 2 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2789 | 1 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4322 ran crn 5677 ‘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-sep 5299 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-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 df-iota 6493 df-fv 6549 |
This theorem is referenced by: pmtrfrn 19321 mrsubrn 34493 mrsub0 34496 mrsubf 34497 mrsubccat 34498 mrsubcn 34499 mrsubco 34501 mrsubvrs 34502 elmsubrn 34508 msubrn 34509 msubf 34512 |
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