Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6979. (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 6384 | . 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | fnex 6979 | . 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) | |
3 | 1, 2 | sylanb 583 | 1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 Vcvv 3494 dom cdm 5554 Fun wfun 6348 Fn wfn 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 |
This theorem is referenced by: opabex 6982 mptexg 6983 mptexgf 6984 funrnex 7654 oprabexd 7675 oprabex 7676 mpoexxg 7772 tfrlem14 8026 hartogslem2 9006 harwdom 9053 abrexexd 30268 mpoexxg2 44385 |
Copyright terms: Public domain | W3C validator |