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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 35595 | . . . . 5 ⊢ Fun Funpart𝐹 | |
2 | funfn 6577 | . . . . 5 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹) | |
3 | 1, 2 | mpbi 229 | . . . 4 ⊢ Funpart𝐹 Fn dom Funpart𝐹 |
4 | 0ex 5302 | . . . . . 6 ⊢ ∅ ∈ V | |
5 | 4 | fconst 6777 | . . . . 5 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} |
6 | ffn 6716 | . . . . 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 469 | . . 3 ⊢ (Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)) |
9 | disjdif 4467 | . . 3 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ | |
10 | fnun 6662 | . . 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 690 | . 2 ⊢ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) |
12 | df-fullfun 35527 | . . . 4 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) | |
13 | 12 | fneq1i 6645 | . . 3 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V) |
14 | unvdif 4470 | . . . . 5 ⊢ (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V | |
15 | 14 | eqcomi 2734 | . . . 4 ⊢ V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) |
16 | 15 | fneq2i 6646 | . . 3 ⊢ ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))) |
17 | 13, 16 | bitri 274 | . 2 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))) |
18 | 11, 17 | mpbir 230 | 1 ⊢ FullFun𝐹 Fn V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 Vcvv 3463 ∖ cdif 3937 ∪ cun 3938 ∩ cin 3939 ∅c0 4318 {csn 4624 × cxp 5670 dom cdm 5672 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 Funpartcfunpart 35501 FullFuncfullfn 35502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-symdif 4237 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-eprel 5576 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 df-fv 6550 df-1st 7989 df-2nd 7990 df-txp 35506 df-singleton 35514 df-singles 35515 df-image 35516 df-funpart 35526 df-fullfun 35527 |
This theorem is referenced by: brfullfun 35600 |
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