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Mirrors > Home > NFE Home > Th. List > fullfunexg | GIF version |
Description: The full function of a set is a set. (Contributed by SF, 9-Mar-2015.) |
Ref | Expression |
---|---|
fullfunexg | ⊢ (F ∈ V → FullFun F ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fullfun 5768 | . 2 ⊢ FullFun F = ((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) | |
2 | idex 5504 | . . . . 5 ⊢ I ∈ V | |
3 | coexg 4749 | . . . . 5 ⊢ (( I ∈ V ∧ F ∈ V) → ( I ∘ F) ∈ V) | |
4 | 2, 3 | mpan 651 | . . . 4 ⊢ (F ∈ V → ( I ∘ F) ∈ V) |
5 | 2 | complex 4104 | . . . . 5 ⊢ ∼ I ∈ V |
6 | coexg 4749 | . . . . 5 ⊢ (( ∼ I ∈ V ∧ F ∈ V) → ( ∼ I ∘ F) ∈ V) | |
7 | 5, 6 | mpan 651 | . . . 4 ⊢ (F ∈ V → ( ∼ I ∘ F) ∈ V) |
8 | difexg 4102 | . . . 4 ⊢ ((( I ∘ F) ∈ V ∧ ( ∼ I ∘ F) ∈ V) → (( I ∘ F) ∖ ( ∼ I ∘ F)) ∈ V) | |
9 | 4, 7, 8 | syl2anc 642 | . . 3 ⊢ (F ∈ V → (( I ∘ F) ∖ ( ∼ I ∘ F)) ∈ V) |
10 | dmexg 5105 | . . . . 5 ⊢ ((( I ∘ F) ∖ ( ∼ I ∘ F)) ∈ V → dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∈ V) | |
11 | complexg 4099 | . . . . 5 ⊢ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∈ V → ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∈ V) | |
12 | 9, 10, 11 | 3syl 18 | . . . 4 ⊢ (F ∈ V → ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∈ V) |
13 | snex 4111 | . . . 4 ⊢ {∅} ∈ V | |
14 | xpexg 5114 | . . . 4 ⊢ (( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∈ V ∧ {∅} ∈ V) → ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ∈ V) | |
15 | 12, 13, 14 | sylancl 643 | . . 3 ⊢ (F ∈ V → ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ∈ V) |
16 | unexg 4101 | . . 3 ⊢ (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∈ V ∧ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ∈ V) → ((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ∈ V) | |
17 | 9, 15, 16 | syl2anc 642 | . 2 ⊢ (F ∈ V → ((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ∈ V) |
18 | 1, 17 | syl5eqel 2437 | 1 ⊢ (F ∈ V → FullFun F ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 ∖ cdif 3206 ∪ cun 3207 ∅c0 3550 {csn 3737 ∘ ccom 4721 I cid 4763 × cxp 4770 dom cdm 4772 FullFun cfullfun 5767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-2nd 4797 df-fullfun 5768 |
This theorem is referenced by: fullfunex 5860 |
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