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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 3777 | . . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅}) | |
3 | 0ex 4790 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | snid 4208 | . . . 4 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3600 | . . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅}) |
6 | 1, 5 | syl6eqel 2709 | . 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})) |
7 | ssun1 3776 | . . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅}) | |
8 | fvprc 6185 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
9 | 8 | con1i 144 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V) |
10 | fvexd 6203 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V) | |
11 | fvbr0 6215 | . . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) | |
12 | 11 | ori 390 | . . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅) |
13 | 12 | con1i 144 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋)) |
14 | brelrng 5355 | . . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹) | |
15 | 9, 10, 13, 14 | syl3anc 1326 | . . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
16 | 7, 15 | sseldi 3601 | . 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})) |
17 | 6, 16 | pm2.61i 176 | 1 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ∅c0 3915 {csn 4177 class class class wbr 4653 ran crn 5115 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 df-iota 5851 df-fv 5896 |
This theorem is referenced by: fvssunirn 6217 dfac4 8945 dfac2 8953 dfacacn 8963 axdc2lem 9270 axcclem 9279 plusffval 17247 staffval 18847 scaffval 18881 lpival 19245 ipffval 19993 nmfval 22393 tchex 23016 tchnmfval 23027 orderseqlem 31749 rrnval 33626 lsatset 34277 fvnonrel 37903 |
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