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| 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 2744 | . . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
| 2 | tz6.12i 6934 | . . . . . . . . . 10 ⊢ (𝑦 ≠ ∅ → ((𝐹‘𝑥) = 𝑦 → 𝑥𝐹𝑦)) | |
| 3 | 1, 2 | biimtrid 242 | . . . . . . . . 9 ⊢ (𝑦 ≠ ∅ → (𝑦 = (𝐹‘𝑥) → 𝑥𝐹𝑦)) | 
| 4 | 3 | eximdv 1917 | . . . . . . . 8 ⊢ (𝑦 ≠ ∅ → (∃𝑥 𝑦 = (𝐹‘𝑥) → ∃𝑥 𝑥𝐹𝑦)) | 
| 5 | vex 3484 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
| 6 | 5 | elrn 5904 | . . . . . . . 8 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦) | 
| 7 | 4, 6 | imbitrrdi 252 | . . . . . . 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 4642 | . . . . 5 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅) | |
| 11 | 9, 10 | imbitrrdi 252 | . . . 4 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (¬ 𝑦 ∈ ran 𝐹 → 𝑦 ∈ {∅})) | 
| 12 | 11 | orrd 864 | . . 3 ⊢ (∃𝑥 𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})) | 
| 13 | 12 | ss2abi 4067 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})} | 
| 14 | df-un 3956 | . 2 ⊢ (ran 𝐹 ∪ {∅}) = {𝑦 ∣ (𝑦 ∈ ran 𝐹 ∨ 𝑦 ∈ {∅})} | |
| 15 | 13, 14 | sseqtrri 4033 | 1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∨ wo 848 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 {csn 4626 class class class wbr 5143 ran crn 5686 ‘cfv 6561 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: fvclex 7983 | 
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