![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4174 | . . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅}) | |
3 | 0ex 5308 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | snid 4665 | . . . 4 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3980 | . . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅}) |
6 | 1, 5 | eqeltrdi 2842 | . 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})) |
7 | ssun1 4173 | . . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅}) | |
8 | fvprc 6884 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
9 | 8 | con1i 147 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V) |
10 | fvexd 6907 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V) | |
11 | fvbr0 6921 | . . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) | |
12 | 11 | ori 860 | . . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅) |
13 | 12 | con1i 147 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋)) |
14 | brelrng 5941 | . . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹) | |
15 | 9, 10, 13, 14 | syl3anc 1372 | . . 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 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3947 ∅c0 4323 {csn 4629 class class class wbr 5149 ran crn 5678 ‘cfv 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-cnv 5685 df-dm 5687 df-rn 5688 df-iota 6496 df-fv 6552 |
This theorem is referenced by: fvn0fvelrn 6923 fvssunirnOLD 6926 orderseqlem 8143 dfac4 10117 dfac2b 10125 dfacacn 10136 axdc2lem 10443 axcclem 10452 seqexw 13982 plusffval 18567 grpsubfval 18868 mulgfval 18952 staffval 20455 scaffval 20490 lpival 20883 ipffval 21201 nmfval 24097 tcphex 24734 tchnmfval 24745 rrnval 36695 lsatset 37860 fvnonrel 42348 |
Copyright terms: Public domain | W3C validator |