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Mirrors > Home > MPE Home > Th. List > fvclss | Structured version Visualization version GIF version |
Description: Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.) |
Ref | Expression |
---|---|
fvclss | ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2828 | . . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
2 | tz6.12i 6690 | . . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ → ((𝐹‘𝑥) = 𝑦 → 𝑥𝐹𝑦)) | |
3 | 1, 2 | syl5bi 244 | . . . . . . . . 9 ⊢ (𝑦 ≠ ∅ → (𝑦 = (𝐹‘𝑥) → 𝑥𝐹𝑦)) |
4 | 3 | eximdv 1914 | . . . . . . . 8 ⊢ (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹‘𝑥) → ∃𝑥 𝑥𝐹𝑦)) |
5 | vex 3497 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
6 | 5 | elrn 5816 | . . . . . . . 8 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦) |
7 | 4, 6 | syl6ibr 254 | . . . . . . 7 ⊢ (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ ran 𝐹)) |
8 | 7 | com12 32 | . . . . . 6 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (𝑦 ≠ ∅ → 𝑦 ∈ ran 𝐹)) |
9 | 8 | necon1bd 3034 | . . . . 5 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (¬ 𝑦 ∈ ran 𝐹 → 𝑦 = ∅)) |
10 | velsn 4576 | . . . . 5 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅) | |
11 | 9, 10 | syl6ibr 254 | . . . 4 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (¬ 𝑦 ∈ ran 𝐹 → 𝑦 ∈ {∅})) |
12 | 11 | orrd 859 | . . 3 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})) |
13 | 12 | ss2abi 4042 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})} |
14 | df-un 3940 | . 2 ⊢ (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})} | |
15 | 13, 14 | sseqtrri 4003 | 1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 {csn 4560 class class class wbr 5058 ran crn 5550 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-cnv 5557 df-dm 5559 df-rn 5560 df-iota 6308 df-fv 6357 |
This theorem is referenced by: fvclex 7654 |
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