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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 2789 | . . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅) |
| 4 | 3 | rneqd 5949 | . 2 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ran ∅) |
| 5 | rn0 5936 | . 2 ⊢ ran ∅ = ∅ | |
| 6 | 4, 5 | eqtrdi 2793 | 1 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ran crn 5686 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: pmtrfrn 19476 mrsubrn 35518 mrsub0 35521 mrsubf 35522 mrsubccat 35523 mrsubcn 35524 mrsubco 35526 mrsubvrs 35527 elmsubrn 35533 msubrn 35534 msubf 35537 |
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