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Mirrors > Home > MPE Home > Th. List > funex | Structured version Visualization version GIF version |
Description: If the domain of a function exists, so does the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 6626. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.) |
Ref | Expression |
---|---|
funex | ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfn 6062 | . 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | fnex 6626 | . 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) | |
3 | 1, 2 | sylanb 564 | 1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 Vcvv 3351 dom cdm 5250 Fun wfun 6026 Fn wfn 6027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 |
This theorem is referenced by: opabex 6628 mptexg 6629 mptexgf 6630 funrnex 7281 oprabexd 7303 oprabex 7304 mpt2exxg 7395 tfrlem14 7641 hartogslem2 8605 harwdom 8652 abrexexd 29686 mpt2exxg2 42645 |
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