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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 6727 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
3 | 1, 2 | eqtrid 2790 | . . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅) |
4 | 3 | rneqd 5821 | . 2 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ran ∅) |
5 | rn0 5809 | . 2 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2795 | 1 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2111 Vcvv 3420 ∅c0 4251 ran crn 5566 ‘cfv 6397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-cnv 5573 df-dm 5575 df-rn 5576 df-iota 6355 df-fv 6405 |
This theorem is referenced by: pmtrfrn 18874 mrsubrn 33211 mrsub0 33214 mrsubf 33215 mrsubccat 33216 mrsubcn 33217 mrsubco 33219 mrsubvrs 33220 elmsubrn 33226 msubrn 33227 msubf 33230 |
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