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Theorem fnex 5297
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 5296. (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 4912 . . 3 (𝐹 Fn A → Rel 𝐹)
21adantr 261 . 2 ((𝐹 Fn A A B) → Rel 𝐹)
3 df-fn 4821 . . 3 (𝐹 Fn A ↔ (Fun 𝐹 dom 𝐹 = A))
4 eleq1a 2083 . . . . . 6 (A B → (dom 𝐹 = A → dom 𝐹 B))
54impcom 116 . . . . 5 ((dom 𝐹 = A A B) → dom 𝐹 B)
6 resfunexg 5296 . . . . 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 4565 . . . 4 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1110eleq1d 2080 . . 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 1224   wcel 1367  Vcvv 2527  dom cdm 4261  cres 4263  Rel wrel 4266  Fun wfun 4812   Fn wfn 4813
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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-coll 3836  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-reu 2283  df-rab 2285  df-v 2529  df-sbc 2734  df-csb 2822  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-uni 3545  df-iun 3623  df-br 3729  df-opab 3783  df-mpt 3784  df-id 3994  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-rn 4272  df-res 4273  df-ima 4274  df-iota 4783  df-fun 4820  df-fn 4821  df-f 4822  df-f1 4823  df-fo 4824  df-f1o 4825  df-fv 4826
This theorem is referenced by:  funex  5298  fex  5302  offval  5631  ofrfval  5632  tfrlemibex  5853
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