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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 6358 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
3 | 1 | fdmi 6052 | . . . 4 ⊢ dom 𝐹 = 𝐴 |
4 | 3 | eleq2i 2693 | . . 3 ⊢ (𝐶 ∈ dom 𝐹 ↔ 𝐶 ∈ 𝐴) |
5 | ndmfv 6218 | . . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅) | |
6 | f0cl.2 | . . . 4 ⊢ ∅ ∈ 𝐵 | |
7 | 5, 6 | syl6eqel 2709 | . . 3 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) ∈ 𝐵) |
8 | 4, 7 | sylnbir 321 | . 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 1990 ∅c0 3915 dom cdm 5114 ⟶wf 5884 ‘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-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-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 |
This theorem is referenced by: harcl 8466 cantnfvalf 8562 rankon 8658 cardon 8770 alephon 8892 ackbij1lem13 9054 ackbij1b 9061 ixxssxr 12187 sadcf 15175 smupf 15200 iccordt 21018 nodense 31842 bdayelon 31892 |
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