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Mirrors > Home > MPE Home > Th. List > fvclex | Structured version Visualization version GIF version |
Description: Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.) |
Ref | Expression |
---|---|
fvclex.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
fvclex | ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvclex.1 | . . . 4 ⊢ 𝐹 ∈ V | |
2 | 1 | rnex 7251 | . . 3 ⊢ ran 𝐹 ∈ V |
3 | p0ex 4985 | . . 3 ⊢ {∅} ∈ V | |
4 | 2, 3 | unex 7107 | . 2 ⊢ (ran 𝐹 ∪ {∅}) ∈ V |
5 | fvclss 6646 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) | |
6 | 4, 5 | ssexi 4938 | 1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∃wex 1852 ∈ wcel 2145 {cab 2757 Vcvv 3351 ∪ cun 3721 ∅c0 4063 {csn 4317 ran crn 5251 ‘cfv 6030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-cnv 5258 df-dm 5260 df-rn 5261 df-iota 5993 df-fv 6038 |
This theorem is referenced by: fvresex 7290 |
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