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Mirrors > Home > MPE Home > Th. List > Mathboxes > fullfunfnv | Structured version Visualization version GIF version |
Description: The full functional part of 𝐹 is a function over V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
fullfunfnv | ⊢ FullFun𝐹 Fn V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funpartfun 33517 | . . . . 5 ⊢ Fun Funpart𝐹 | |
2 | funfn 6354 | . . . . 5 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹) | |
3 | 1, 2 | mpbi 233 | . . . 4 ⊢ Funpart𝐹 Fn dom Funpart𝐹 |
4 | 0ex 5175 | . . . . . 6 ⊢ ∅ ∈ V | |
5 | 4 | fconst 6539 | . . . . 5 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} |
6 | ffn 6487 | . . . . 5 ⊢ (((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} → ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹) |
8 | 3, 7 | pm3.2i 474 | . . 3 ⊢ (Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)) |
9 | disjdif 4379 | . . 3 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ | |
10 | fnun 6434 | . . 3 ⊢ (((Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)) ∧ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅) → (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))) | |
11 | 8, 9, 10 | mp2an 691 | . 2 ⊢ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) |
12 | df-fullfun 33449 | . . . 4 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) | |
13 | 12 | fneq1i 6420 | . . 3 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V) |
14 | unvdif 4381 | . . . . 5 ⊢ (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V | |
15 | 14 | eqcomi 2807 | . . . 4 ⊢ V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) |
16 | 15 | fneq2i 6421 | . . 3 ⊢ ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))) |
17 | 13, 16 | bitri 278 | . 2 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))) |
18 | 11, 17 | mpbir 234 | 1 ⊢ FullFun𝐹 Fn V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 {csn 4525 × cxp 5517 dom cdm 5519 Fun wfun 6318 Fn wfn 6319 ⟶wf 6320 Funpartcfunpart 33423 FullFuncfullfn 33424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-symdif 4169 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-eprel 5430 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-1st 7671 df-2nd 7672 df-txp 33428 df-singleton 33436 df-singles 33437 df-image 33438 df-funpart 33448 df-fullfun 33449 |
This theorem is referenced by: brfullfun 33522 |
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