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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 35944 | . . . . 5 ⊢ Fun Funpart𝐹 | |
| 2 | funfn 6596 | . . . . 5 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹) | |
| 3 | 1, 2 | mpbi 230 | . . . 4 ⊢ Funpart𝐹 Fn dom Funpart𝐹 |
| 4 | 0ex 5307 | . . . . . 6 ⊢ ∅ ∈ V | |
| 5 | 4 | fconst 6794 | . . . . 5 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} |
| 6 | ffn 6736 | . . . . 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 470 | . . 3 ⊢ (Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)) |
| 9 | disjdif 4472 | . . 3 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ | |
| 10 | fnun 6682 | . . 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 692 | . 2 ⊢ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) |
| 12 | df-fullfun 35876 | . . . 4 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) | |
| 13 | 12 | fneq1i 6665 | . . 3 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V) |
| 14 | unvdif 4475 | . . . . 5 ⊢ (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V | |
| 15 | 14 | eqcomi 2746 | . . . 4 ⊢ V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) |
| 16 | 15 | fneq2i 6666 | . . 3 ⊢ ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))) |
| 17 | 13, 16 | bitri 275 | . 2 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))) |
| 18 | 11, 17 | mpbir 231 | 1 ⊢ FullFun𝐹 Fn V |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 × cxp 5683 dom cdm 5685 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 Funpartcfunpart 35850 FullFuncfullfn 35851 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-symdif 4253 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 df-singleton 35863 df-singles 35864 df-image 35865 df-funpart 35875 df-fullfun 35876 |
| This theorem is referenced by: brfullfun 35949 |
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