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| Mirrors > Home > MPE Home > Th. List > fvrn0 | Structured version Visualization version GIF version | ||
| Description: A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.) |
| Ref | Expression |
|---|---|
| fvrn0 | ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . . 3 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) = ∅) | |
| 2 | ssun2 4179 | . . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅}) | |
| 3 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 4 | 3 | snid 4662 | . . . 4 ⊢ ∅ ∈ {∅} |
| 5 | 2, 4 | sselii 3980 | . . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅}) |
| 6 | 1, 5 | eqeltrdi 2849 | . 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})) |
| 7 | ssun1 4178 | . . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅}) | |
| 8 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
| 9 | 8 | con1i 147 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V) |
| 10 | fvexd 6921 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V) | |
| 11 | fvbr0 6935 | . . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) | |
| 12 | 11 | ori 862 | . . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅) |
| 13 | 12 | con1i 147 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋)) |
| 14 | brelrng 5952 | . . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹) | |
| 15 | 9, 10, 13, 14 | syl3anc 1373 | . . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
| 16 | 7, 15 | sselid 3981 | . 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})) |
| 17 | 6, 16 | pm2.61i 182 | 1 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∅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: fvn0fvelrn 6937 fvssunirnOLD 6940 orderseqlem 8182 dfac4 10162 dfac2b 10171 dfacacn 10182 axdc2lem 10488 axcclem 10497 seqexw 14058 plusffval 18659 grpsubfval 19001 mulgfval 19087 staffval 20842 scaffval 20878 lpival 21334 ipffval 21666 nmfval 24601 tcphex 25251 tchnmfval 25262 rrnval 37834 lsatset 38991 fvnonrel 43610 |
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