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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 4149 | . . . 4 ⊢ {∅} ⊆ (ran 𝐹 ∪ {∅}) | |
3 | 0ex 5211 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | snid 4601 | . . . 4 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3964 | . . 3 ⊢ ∅ ∈ (ran 𝐹 ∪ {∅}) |
6 | 1, 5 | eqeltrdi 2921 | . 2 ⊢ ((𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})) |
7 | ssun1 4148 | . . 3 ⊢ ran 𝐹 ⊆ (ran 𝐹 ∪ {∅}) | |
8 | fvprc 6663 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
9 | 8 | con1i 149 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ V) |
10 | fvexd 6685 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ V) | |
11 | fvbr0 6697 | . . . . . 6 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) | |
12 | 11 | ori 857 | . . . . 5 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅) |
13 | 12 | con1i 149 | . . . 4 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋𝐹(𝐹‘𝑋)) |
14 | brelrng 5811 | . . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V ∧ 𝑋𝐹(𝐹‘𝑋)) → (𝐹‘𝑋) ∈ ran 𝐹) | |
15 | 9, 10, 13, 14 | syl3anc 1367 | . . 3 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ ran 𝐹) |
16 | 7, 15 | sseldi 3965 | . 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})) |
17 | 6, 16 | pm2.61i 184 | 1 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∅c0 4291 {csn 4567 class class class wbr 5066 ran crn 5556 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 df-iota 6314 df-fv 6363 |
This theorem is referenced by: fvssunirn 6699 dfac4 9548 dfac2b 9556 dfacacn 9567 axdc2lem 9870 axcclem 9879 seqexw 13386 plusffval 17858 grpsubfval 18147 mulgfval 18226 staffval 19618 scaffval 19652 lpival 20018 ipffval 20792 nmfval 23198 tcphex 23820 tchnmfval 23831 orderseqlem 33094 rrnval 35120 lsatset 36141 fvnonrel 39977 |
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