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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 6898 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
3 | 1, 2 | eqtrid 2786 | . . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅) |
4 | 3 | rneqd 5951 | . 2 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ran ∅) |
5 | rn0 5938 | . 2 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2790 | 1 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 ran crn 5689 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-cnv 5696 df-dm 5698 df-rn 5699 df-iota 6515 df-fv 6570 |
This theorem is referenced by: pmtrfrn 19490 mrsubrn 35497 mrsub0 35500 mrsubf 35501 mrsubccat 35502 mrsubcn 35503 mrsubco 35505 mrsubvrs 35506 elmsubrn 35512 msubrn 35513 msubf 35516 |
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