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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 4175 | . . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅}) | |
3 | 0ex 5311 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | snid 4669 | . . . 4 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3979 | . . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅}) |
6 | 1, 5 | eqeltrdi 2837 | . 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})) |
7 | ssun1 4174 | . . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅}) | |
8 | fvprc 6894 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
9 | 8 | con1i 147 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V) |
10 | fvexd 6917 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V) | |
11 | fvbr0 6931 | . . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) | |
12 | 11 | ori 859 | . . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅) |
13 | 12 | con1i 147 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋)) |
14 | brelrng 5947 | . . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹) | |
15 | 9, 10, 13, 14 | syl3anc 1368 | . . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
16 | 7, 15 | sselid 3980 | . 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 1533 ∈ wcel 2098 Vcvv 3473 ∪ cun 3947 ∅c0 4326 {csn 4632 class class class wbr 5152 ran crn 5683 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-cnv 5690 df-dm 5692 df-rn 5693 df-iota 6505 df-fv 6561 |
This theorem is referenced by: fvn0fvelrn 6933 fvssunirnOLD 6936 orderseqlem 8168 dfac4 10153 dfac2b 10161 dfacacn 10172 axdc2lem 10479 axcclem 10488 seqexw 14022 plusffval 18613 grpsubfval 18947 mulgfval 19032 staffval 20734 scaffval 20770 lpival 21221 ipffval 21587 nmfval 24517 tcphex 25165 tchnmfval 25176 rrnval 37333 lsatset 38494 fvnonrel 43058 |
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