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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 6656 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
3 | 1, 2 | syl5eq 2865 | . . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅) |
4 | 3 | rneqd 5801 | . 2 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ran ∅) |
5 | rn0 5789 | . 2 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | syl6eq 2869 | 1 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 ran crn 5549 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 df-iota 6307 df-fv 6356 |
This theorem is referenced by: pmtrfrn 18515 mrsubrn 32657 mrsub0 32660 mrsubf 32661 mrsubccat 32662 mrsubcn 32663 mrsubco 32665 mrsubvrs 32666 elmsubrn 32672 msubrn 32673 msubf 32676 |
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