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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 7115 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
3 | 1 | fdmi 6757 | . . . 4 ⊢ dom 𝐹 = 𝐴 |
4 | 3 | eleq2i 2830 | . . 3 ⊢ (𝐶 ∈ dom 𝐹 ↔ 𝐶 ∈ 𝐴) |
5 | ndmfv 6954 | . . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅) | |
6 | f0cl.2 | . . . 4 ⊢ ∅ ∈ 𝐵 | |
7 | 5, 6 | eqeltrdi 2846 | . . 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 2103 ∅c0 4347 dom cdm 5699 ⟶wf 6568 ‘cfv 6572 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-fv 6580 |
This theorem is referenced by: harcl 9624 cantnfvalf 9730 rankon 9860 cardon 10009 alephon 10134 ackbij1lem13 10296 ackbij1b 10303 ixxssxr 13415 sadcf 16493 smupf 16518 iccordt 23236 nodense 27746 bdayelon 27830 madessno 27908 oldssno 27909 newssno 27910 |
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