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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 2767 | . . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
2 | tz6.12i 6376 | . . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ → ((𝐹‘𝑥) = 𝑦 → 𝑥𝐹𝑦)) | |
3 | 1, 2 | syl5bi 232 | . . . . . . . . 9 ⊢ (𝑦 ≠ ∅ → (𝑦 = (𝐹‘𝑥) → 𝑥𝐹𝑦)) |
4 | 3 | eximdv 1995 | . . . . . . . 8 ⊢ (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹‘𝑥) → ∃𝑥 𝑥𝐹𝑦)) |
5 | vex 3343 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
6 | 5 | elrn 5521 | . . . . . . . 8 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦) |
7 | 4, 6 | syl6ibr 242 | . . . . . . 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 2950 | . . . . 5 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (¬ 𝑦 ∈ ran 𝐹 → 𝑦 = ∅)) |
10 | velsn 4337 | . . . . 5 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅) | |
11 | 9, 10 | syl6ibr 242 | . . . 4 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (¬ 𝑦 ∈ ran 𝐹 → 𝑦 ∈ {∅})) |
12 | 11 | orrd 392 | . . 3 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})) |
13 | 12 | ss2abi 3815 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})} |
14 | df-un 3720 | . 2 ⊢ (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})} | |
15 | 13, 14 | sseqtr4i 3779 | 1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 = wceq 1632 ∃wex 1853 ∈ wcel 2139 {cab 2746 ≠ wne 2932 ∪ cun 3713 ⊆ wss 3715 ∅c0 4058 {csn 4321 class class class wbr 4804 ran crn 5267 ‘cfv 6049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-cnv 5274 df-dm 5276 df-rn 5277 df-iota 6012 df-fv 6057 |
This theorem is referenced by: fvclex 7304 |
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