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Theorem fnex 5326
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 5325. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fnex ((𝐹 Fn A A B) → 𝐹 V)

Proof of Theorem fnex
StepHypRef Expression
1 fnrel 4940 . . 3 (𝐹 Fn A → Rel 𝐹)
21adantr 261 . 2 ((𝐹 Fn A A B) → Rel 𝐹)
3 df-fn 4848 . . 3 (𝐹 Fn A ↔ (Fun 𝐹 dom 𝐹 = A))
4 eleq1a 2106 . . . . . 6 (A B → (dom 𝐹 = A → dom 𝐹 B))
54impcom 116 . . . . 5 ((dom 𝐹 = A A B) → dom 𝐹 B)
6 resfunexg 5325 . . . . 5 ((Fun 𝐹 dom 𝐹 B) → (𝐹 ↾ dom 𝐹) V)
75, 6sylan2 270 . . . 4 ((Fun 𝐹 (dom 𝐹 = A A B)) → (𝐹 ↾ dom 𝐹) V)
87anassrs 380 . . 3 (((Fun 𝐹 dom 𝐹 = A) A B) → (𝐹 ↾ dom 𝐹) V)
93, 8sylanb 268 . 2 ((𝐹 Fn A A B) → (𝐹 ↾ dom 𝐹) V)
10 resdm 4592 . . . 4 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1110eleq1d 2103 . . 3 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) V ↔ 𝐹 V))
1211biimpa 280 . 2 ((Rel 𝐹 (𝐹 ↾ dom 𝐹) V) → 𝐹 V)
132, 9, 12syl2anc 391 1 ((𝐹 Fn A A B) → 𝐹 V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  dom cdm 4288  cres 4290  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by:  funex  5327  fex  5331  offval  5661  ofrfval  5662  tfrlemibex  5884  fndmeng  6225  frecfzennn  8844
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