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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 7222. (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 6579 | . 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | fnex 7222 | . 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) | |
3 | 1, 2 | sylanb 579 | 1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2104 Vcvv 3472 dom cdm 5677 Fun wfun 6538 Fn wfn 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 |
This theorem is referenced by: opabex 7225 mptexg 7226 mptexgf 7227 funrnex 7944 oprabexd 7966 oprabex 7967 mpoexxg 8066 tfrlem14 8395 hartogslem2 9542 harwdom 9590 abrexexd 32011 mpoexxg2 47103 |
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