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Mirrors > Home > ILE Home > Th. List > fnex | 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 5382. (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 4997 | . . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | 1 | adantr 261 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → Rel 𝐹) |
3 | df-fn 4905 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
4 | eleq1a 2109 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵)) | |
5 | 4 | impcom 116 | . . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵) |
6 | resfunexg 5382 | . . . . 5 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) | |
7 | 5, 6 | sylan2 270 | . . . 4 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V) |
8 | 7 | anassrs 380 | . . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
9 | 3, 8 | sylanb 268 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
10 | resdm 4649 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | |
11 | 10 | eleq1d 2106 | . . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V)) |
12 | 11 | biimpa 280 | . 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V) |
13 | 2, 9, 12 | syl2anc 391 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 dom cdm 4345 ↾ cres 4347 Rel wrel 4350 Fun wfun 4896 Fn wfn 4897 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 |
This theorem is referenced by: funex 5384 fex 5388 offval 5719 ofrfval 5720 tfrlemibex 5943 fndmeng 6289 frecfzennn 9203 |
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