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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 7212. See fnexALT 7932 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 6648 | . 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | df-fn 6543 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
3 | eleq1a 2829 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵)) | |
4 | 3 | impcom 409 | . . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵) |
5 | resfunexg 7212 | . . . . 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 469 | . . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
8 | 2, 7 | sylanb 582 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
9 | resdm 6024 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | |
10 | 9 | eleq1d 2819 | . . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V)) |
11 | 10 | biimpa 478 | . 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V) |
12 | 1, 8, 11 | syl2an2r 684 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 dom cdm 5675 ↾ cres 5677 Rel wrel 5680 Fun wfun 6534 Fn wfn 6535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 |
This theorem is referenced by: fnexd 7215 funex 7216 fex 7223 offval 7674 fndmexb 7894 suppvalfn 8149 suppfnss 8169 fnsuppeq0 8172 wfrlem15OLD 8318 fndmeng 9031 fdmfifsupp 9369 cfsmolem 10261 axcc2lem 10427 unirnfdomd 10558 prdsbas2 17411 prdsplusgval 17415 prdsmulrval 17417 prdsleval 17419 prdsdsval 17420 prdsvscaval 17421 xpscf 17507 brssc 17757 sscpwex 17758 ssclem 17762 isssc 17763 rescval2 17771 reschom 17774 rescabsOLD 17779 isfuncd 17811 dprdw 19872 prdsmgp 20122 dsmmbas2 21276 dsmmelbas 21278 ptval 23056 prdstopn 23114 qtoptop 23186 imastopn 23206 fnpreimac 31874 suppss3 31927 ofcfval 33034 dya2iocuni 33220 tfsconcatun 42020 stoweidlem27 44678 stoweidlem59 44710 omeiunle 45168 preimafvelsetpreimafv 45991 fundcmpsurinjlem2 46002 |
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