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Mirrors > Home > MPE Home > Th. List > f0cli | Structured version Visualization version GIF version |
Description: Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.) |
Ref | Expression |
---|---|
f0cl.1 | ⊢ 𝐹:𝐴⟶𝐵 |
f0cl.2 | ⊢ ∅ ∈ 𝐵 |
Ref | Expression |
---|---|
f0cli | ⊢ (𝐹‘𝐶) ∈ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0cl.1 | . . 3 ⊢ 𝐹:𝐴⟶𝐵 | |
2 | 1 | ffvelrni 6521 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
3 | 1 | fdmi 6213 | . . . 4 ⊢ dom 𝐹 = 𝐴 |
4 | 3 | eleq2i 2831 | . . 3 ⊢ (𝐶 ∈ dom 𝐹 ↔ 𝐶 ∈ 𝐴) |
5 | ndmfv 6379 | . . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅) | |
6 | f0cl.2 | . . . 4 ⊢ ∅ ∈ 𝐵 | |
7 | 5, 6 | syl6eqel 2847 | . . 3 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) ∈ 𝐵) |
8 | 4, 7 | sylnbir 320 | . 2 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
9 | 2, 8 | pm2.61i 176 | 1 ⊢ (𝐹‘𝐶) ∈ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2139 ∅c0 4058 dom cdm 5266 ⟶wf 6045 ‘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-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-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-fv 6057 |
This theorem is referenced by: harcl 8631 cantnfvalf 8735 rankon 8831 cardon 8960 alephon 9082 ackbij1lem13 9246 ackbij1b 9253 ixxssxr 12380 sadcf 15377 smupf 15402 iccordt 21220 nodense 32148 bdayelon 32198 |
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