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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 codomain 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 | ffvelcdmi 7119 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
3 | 1 | fdmi 6760 | . . . 4 ⊢ dom 𝐹 = 𝐴 |
4 | 3 | eleq2i 2836 | . . 3 ⊢ (𝐶 ∈ dom 𝐹 ↔ 𝐶 ∈ 𝐴) |
5 | ndmfv 6957 | . . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅) | |
6 | f0cl.2 | . . . 4 ⊢ ∅ ∈ 𝐵 | |
7 | 5, 6 | eqeltrdi 2852 | . . 3 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) ∈ 𝐵) |
8 | 4, 7 | sylnbir 331 | . 2 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
9 | 2, 8 | pm2.61i 182 | 1 ⊢ (𝐹‘𝐶) ∈ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2108 ∅c0 4352 dom cdm 5700 ⟶wf 6571 ‘cfv 6575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-fv 6583 |
This theorem is referenced by: harcl 9630 cantnfvalf 9736 rankon 9866 cardon 10015 alephon 10140 ackbij1lem13 10302 ackbij1b 10309 ixxssxr 13421 sadcf 16501 smupf 16526 iccordt 23245 nodense 27757 bdayelon 27841 madessno 27919 oldssno 27920 newssno 27921 |
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