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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 3920 | . . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅}) | |
3 | 0ex 4942 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | snid 4353 | . . . 4 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3741 | . . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅}) |
6 | 1, 5 | syl6eqel 2847 | . 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})) |
7 | ssun1 3919 | . . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅}) | |
8 | fvprc 6347 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
9 | 8 | con1i 144 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V) |
10 | fvexd 6365 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V) | |
11 | fvbr0 6377 | . . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) | |
12 | 11 | ori 389 | . . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅) |
13 | 12 | con1i 144 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋)) |
14 | brelrng 5510 | . . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹) | |
15 | 9, 10, 13, 14 | syl3anc 1477 | . . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
16 | 7, 15 | sseldi 3742 | . 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 1632 ∈ wcel 2139 Vcvv 3340 ∪ cun 3713 ∅c0 4058 {csn 4321 class class class wbr 4804 ran crn 5267 ‘cfv 6049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-cnv 5274 df-dm 5276 df-rn 5277 df-iota 6012 df-fv 6057 |
This theorem is referenced by: fvssunirn 6379 dfac4 9155 dfac2b 9163 dfac2OLD 9165 dfacacn 9175 axdc2lem 9482 axcclem 9491 plusffval 17468 staffval 19069 scaffval 19103 lpival 19467 ipffval 20215 nmfval 22614 tchex 23236 tchnmfval 23247 orderseqlem 32079 rrnval 33957 lsatset 34798 fvnonrel 38423 |
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