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Theorem fnex 5718
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 5717. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fnex ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Proof of Theorem fnex
StepHypRef Expression
1 fnrel 5296 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 274 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → Rel 𝐹)
3 df-fn 5201 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
4 eleq1a 2242 . . . . . 6 (𝐴𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹𝐵))
54impcom 124 . . . . 5 ((dom 𝐹 = 𝐴𝐴𝐵) → dom 𝐹𝐵)
6 resfunexg 5717 . . . . 5 ((Fun 𝐹 ∧ dom 𝐹𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
75, 6sylan2 284 . . . 4 ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴𝐴𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
87anassrs 398 . . 3 (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
93, 8sylanb 282 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
10 resdm 4930 . . . 4 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1110eleq1d 2239 . . 3 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
1211biimpa 294 . 2 ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
132, 9, 12syl2anc 409 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  Vcvv 2730  dom cdm 4611  cres 4613  Rel wrel 4616  Fun wfun 5192   Fn wfn 5193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206
This theorem is referenced by:  funex  5719  fex  5725  offval  6068  ofrfval  6069  tfrlemibex  6308  tfr1onlembex  6324  fndmeng  6788  cc2lem  7228  frecfzennn  10382
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