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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 6266 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
3 | 1 | fdmi 5965 | . . . 4 ⊢ dom 𝐹 = 𝐴 |
4 | 3 | eleq2i 2680 | . . 3 ⊢ (𝐶 ∈ dom 𝐹 ↔ 𝐶 ∈ 𝐴) |
5 | ndmfv 6128 | . . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅) | |
6 | f0cl.2 | . . . 4 ⊢ ∅ ∈ 𝐵 | |
7 | 5, 6 | syl6eqel 2696 | . . 3 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) ∈ 𝐵) |
8 | 4, 7 | sylnbir 320 | . 2 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
9 | 2, 8 | pm2.61i 175 | 1 ⊢ (𝐹‘𝐶) ∈ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 1977 ∅c0 3874 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: harcl 8349 cantnfvalf 8445 rankon 8541 cardon 8653 alephon 8775 ackbij1lem13 8937 ackbij1b 8944 ixxssxr 12058 sadcf 15013 smupf 15038 iccordt 20828 |
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