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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 7620 | . . 3 ⊢ ran 𝐹 ∈ V |
3 | snex 5335 | . . 3 ⊢ {∅} ∈ V | |
4 | 2, 3 | unex 7472 | . 2 ⊢ (ran 𝐹 ∪ {∅}) ∈ V |
5 | fvclss 7004 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) | |
6 | 4, 5 | ssexi 5229 | 1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∃wex 1779 ∈ wcel 2113 {cab 2802 Vcvv 3497 ∪ cun 3937 ∅c0 4294 {csn 4570 ran crn 5559 ‘cfv 6358 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-cnv 5566 df-dm 5568 df-rn 5569 df-iota 6317 df-fv 6366 |
This theorem is referenced by: fvresex 7664 |
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