![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2739 | . . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
2 | tz6.12i 6919 | . . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ → ((𝐹‘𝑥) = 𝑦 → 𝑥𝐹𝑦)) | |
3 | 1, 2 | biimtrid 241 | . . . . . . . . 9 ⊢ (𝑦 ≠ ∅ → (𝑦 = (𝐹‘𝑥) → 𝑥𝐹𝑦)) |
4 | 3 | eximdv 1920 | . . . . . . . 8 ⊢ (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹‘𝑥) → ∃𝑥 𝑥𝐹𝑦)) |
5 | vex 3478 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
6 | 5 | elrn 5893 | . . . . . . . 8 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦) |
7 | 4, 6 | syl6ibr 251 | . . . . . . 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 2958 | . . . . 5 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (¬ 𝑦 ∈ ran 𝐹 → 𝑦 = ∅)) |
10 | velsn 4644 | . . . . 5 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅) | |
11 | 9, 10 | syl6ibr 251 | . . . 4 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (¬ 𝑦 ∈ ran 𝐹 → 𝑦 ∈ {∅})) |
12 | 11 | orrd 861 | . . 3 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})) |
13 | 12 | ss2abi 4063 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})} |
14 | df-un 3953 | . 2 ⊢ (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})} | |
15 | 13, 14 | sseqtrri 4019 | 1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 845 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2709 ≠ wne 2940 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 {csn 4628 class class class wbr 5148 ran crn 5677 ‘cfv 6543 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 df-iota 6495 df-fv 6551 |
This theorem is referenced by: fvclex 7944 |
Copyright terms: Public domain | W3C validator |