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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 7381 | . . 3 ⊢ ran 𝐹 ∈ V |
3 | snex 5142 | . . 3 ⊢ {∅} ∈ V | |
4 | 2, 3 | unex 7235 | . 2 ⊢ (ran 𝐹 ∪ {∅}) ∈ V |
5 | fvclss 6774 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) | |
6 | 4, 5 | ssexi 5042 | 1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∃wex 1823 ∈ wcel 2107 {cab 2763 Vcvv 3398 ∪ cun 3790 ∅c0 4141 {csn 4398 ran crn 5358 ‘cfv 6137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-cnv 5365 df-dm 5367 df-rn 5368 df-iota 6101 df-fv 6145 |
This theorem is referenced by: fvresex 7420 |
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