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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 33408 | . . . . 5 ⊢ Fun Funpart𝐹 | |
2 | funfn 6388 | . . . . 5 ⊢ (Fun Funpart𝐹 ↔ Funpart𝐹 Fn dom Funpart𝐹) | |
3 | 1, 2 | mpbi 232 | . . . 4 ⊢ Funpart𝐹 Fn dom Funpart𝐹 |
4 | 0ex 5214 | . . . . . 6 ⊢ ∅ ∈ V | |
5 | 4 | fconst 6568 | . . . . 5 ⊢ ((V ∖ dom Funpart𝐹) × {∅}):(V ∖ dom Funpart𝐹)⟶{∅} |
6 | ffn 6517 | . . . . 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 473 | . . 3 ⊢ (Funpart𝐹 Fn dom Funpart𝐹 ∧ ((V ∖ dom Funpart𝐹) × {∅}) Fn (V ∖ dom Funpart𝐹)) |
9 | disjdif 4424 | . . 3 ⊢ (dom Funpart𝐹 ∩ (V ∖ dom Funpart𝐹)) = ∅ | |
10 | fnun 6466 | . . 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 33340 | . . . 4 ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) | |
13 | 12 | fneq1i 6453 | . . 3 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V) |
14 | unvdif 4426 | . . . . 5 ⊢ (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) = V | |
15 | 14 | eqcomi 2833 | . . . 4 ⊢ V = (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹)) |
16 | 15 | fneq2i 6454 | . . 3 ⊢ ((Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))) |
17 | 13, 16 | bitri 277 | . 2 ⊢ (FullFun𝐹 Fn V ↔ (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅})) Fn (dom Funpart𝐹 ∪ (V ∖ dom Funpart𝐹))) |
18 | 11, 17 | mpbir 233 | 1 ⊢ FullFun𝐹 Fn V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 Vcvv 3497 ∖ cdif 3936 ∪ cun 3937 ∩ cin 3938 ∅c0 4294 {csn 4570 × cxp 5556 dom cdm 5558 Fun wfun 6352 Fn wfn 6353 ⟶wf 6354 Funpartcfunpart 33314 FullFuncfullfn 33315 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-symdif 4222 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-eprel 5468 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-1st 7692 df-2nd 7693 df-txp 33319 df-singleton 33327 df-singles 33328 df-image 33329 df-funpart 33339 df-fullfun 33340 |
This theorem is referenced by: brfullfun 33413 |
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