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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 6638 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
3 | 1, 2 | syl5eq 2845 | . . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅) |
4 | 3 | rneqd 5772 | . 2 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ran ∅) |
5 | rn0 5760 | . 2 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2849 | 1 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 ran crn 5520 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 df-iota 6283 df-fv 6332 |
This theorem is referenced by: pmtrfrn 18578 mrsubrn 32873 mrsub0 32876 mrsubf 32877 mrsubccat 32878 mrsubcn 32879 mrsubco 32881 mrsubvrs 32882 elmsubrn 32888 msubrn 32889 msubf 32892 |
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