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Mirrors > Home > MPE Home > Th. List > fnex | Structured version Visualization version GIF version |
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 6972. See fnexALT 7646 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fnex | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 6448 | . 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | df-fn 6352 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
3 | eleq1a 2908 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵)) | |
4 | 3 | impcom 410 | . . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵) |
5 | resfunexg 6972 | . . . . 5 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) | |
6 | 4, 5 | sylan2 594 | . . . 4 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V) |
7 | 6 | anassrs 470 | . . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
8 | 2, 7 | sylanb 583 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
9 | resdm 5891 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | |
10 | 9 | eleq1d 2897 | . . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V)) |
11 | 10 | biimpa 479 | . 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V) |
12 | 1, 8, 11 | syl2an2r 683 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 dom cdm 5549 ↾ cres 5551 Rel wrel 5554 Fun wfun 6343 Fn wfn 6344 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: fnexd 6975 funex 6976 fex 6983 offval 7410 ofrfval 7411 suppvalfn 7831 suppfnss 7849 fnsuppeq0 7852 wfrlem15 7963 fndmeng 8581 fdmfifsupp 8837 cfsmolem 9686 axcc2lem 9852 unirnfdomd 9983 prdsbas2 16736 prdsplusgval 16740 prdsmulrval 16742 prdsleval 16744 prdsdsval 16745 prdsvscaval 16746 xpscf 16832 brssc 17078 sscpwex 17079 ssclem 17083 isssc 17084 rescval2 17092 reschom 17094 rescabs 17097 isfuncd 17129 dprdw 19126 prdsmgp 19354 dsmmbas2 20875 dsmmelbas 20877 ptval 22172 elptr 22175 prdstopn 22230 qtoptop 22302 imastopn 22322 fnpreimac 30410 suppss3 30454 ofcfval 31352 dya2iocuni 31536 trpredex 33071 stoweidlem27 42306 stoweidlem59 42338 omeiunle 42793 preimafvelsetpreimafv 43542 fundcmpsurinjlem2 43553 |
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