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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 7841 | . . 3 ⊢ ran 𝐹 ∈ V |
3 | snex 5386 | . . 3 ⊢ {∅} ∈ V | |
4 | 2, 3 | unex 7672 | . 2 ⊢ (ran 𝐹 ∪ {∅}) ∈ V |
5 | fvclss 7185 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) | |
6 | 4, 5 | ssexi 5277 | 1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2714 Vcvv 3443 ∪ cun 3906 ∅c0 4280 {csn 4584 ran crn 5632 ‘cfv 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-cnv 5639 df-dm 5641 df-rn 5642 df-iota 6445 df-fv 6501 |
This theorem is referenced by: fvresex 7884 |
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