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Mirrors > Home > MPE Home > Th. List > fnex | Structured version Visualization version GIF version |
Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 6751. See fnexALT 7411 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fnex | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 6234 | . 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | df-fn 6138 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
3 | eleq1a 2853 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵)) | |
4 | 3 | impcom 398 | . . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵) |
5 | resfunexg 6751 | . . . . 5 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) | |
6 | 4, 5 | sylan2 586 | . . . 4 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V) |
7 | 6 | anassrs 461 | . . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
8 | 2, 7 | sylanb 576 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
9 | resdm 5691 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | |
10 | 9 | eleq1d 2843 | . . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V)) |
11 | 10 | biimpa 470 | . 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V) |
12 | 1, 8, 11 | syl2an2r 675 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 Vcvv 3397 dom cdm 5355 ↾ cres 5357 Rel wrel 5360 Fun wfun 6129 Fn wfn 6130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 |
This theorem is referenced by: funex 6754 fex 6761 offval 7181 ofrfval 7182 suppvalfn 7583 suppfnss 7601 suppfnssOLD 7602 fnsuppeq0 7605 wfrlem15 7712 fndmeng 8319 fdmfifsupp 8573 cfsmolem 9427 axcc2lem 9593 unirnfdomd 9724 prdsbas2 16515 prdsplusgval 16519 prdsmulrval 16521 prdsleval 16523 prdsdsval 16524 prdsvscaval 16525 brssc 16859 sscpwex 16860 ssclem 16864 isssc 16865 rescval2 16873 reschom 16875 rescabs 16878 isfuncd 16910 dprdw 18796 prdsmgp 18997 dsmmbas2 20480 dsmmelbas 20482 ptval 21782 elptr 21785 prdstopn 21840 qtoptop 21912 imastopn 21932 fnpreimac 30050 suppss3 30082 ofcfval 30772 dya2iocuni 30957 trpredex 32339 fnexd 40241 stoweidlem27 41163 stoweidlem59 41195 omeiunle 41650 |
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