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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 7235. See fnexALT 7975 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 6670 | . 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
| 2 | df-fn 6564 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
| 3 | eleq1a 2836 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵)) | |
| 4 | 3 | impcom 407 | . . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵) |
| 5 | resfunexg 7235 | . . . . 5 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) | |
| 6 | 4, 5 | sylan2 593 | . . . 4 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V) |
| 7 | 6 | anassrs 467 | . . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
| 8 | 2, 7 | sylanb 581 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V) |
| 9 | resdm 6044 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) | |
| 10 | 9 | eleq1d 2826 | . . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V)) |
| 11 | 10 | biimpa 476 | . 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V) |
| 12 | 1, 8, 11 | syl2an2r 685 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 Fun wfun 6555 Fn wfn 6556 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 |
| This theorem is referenced by: fnexd 7238 funex 7239 fex 7246 offval 7706 fndmexb 7928 suppvalfn 8193 suppfnss 8214 fnsuppeq0 8217 wfrlem15OLD 8363 fndmeng 9075 fdmfifsupp 9415 cfsmolem 10310 axcc2lem 10476 unirnfdomd 10607 prdsbas2 17514 prdsplusgval 17518 prdsmulrval 17520 prdsleval 17522 prdsdsval 17523 prdsvscaval 17524 xpscf 17610 brssc 17858 sscpwex 17859 ssclem 17863 isssc 17864 rescval2 17872 reschom 17874 rescabsOLD 17878 isfuncd 17910 dprdw 20030 prdsmgp 20148 dsmmbas2 21757 dsmmelbas 21759 ptval 23578 prdstopn 23636 qtoptop 23708 imastopn 23728 fnpreimac 32681 suppss3 32735 ofcfval 34099 dya2iocuni 34285 tfsconcatun 43350 stoweidlem27 46042 stoweidlem59 46074 omeiunle 46532 preimafvelsetpreimafv 47375 fundcmpsurinjlem2 47386 |
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