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Mirrors > Home > MPE Home > Th. List > fmpo | Structured version Visualization version GIF version |
Description: Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
Ref | Expression |
---|---|
fmpo.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
fmpo | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmpo.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | fmpox 7765 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷) |
3 | iunxpconst 5624 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) | |
4 | 3 | feq2i 6506 | . 2 ⊢ (𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷) |
5 | 2, 4 | bitri 277 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {csn 4567 ∪ ciun 4919 × cxp 5553 ⟶wf 6351 ∈ cmpo 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: fnmpo 7767 ovmpoelrn 7770 fmpoco 7790 eroprf 8395 omxpenlem 8618 mapxpen 8683 dffi3 8895 ixpiunwdom 9055 cantnfvalf 9128 iunfictbso 9540 axdc4lem 9877 axcclem 9879 addpqf 10366 mulpqf 10368 subf 10888 xaddf 12618 xmulf 12666 ixxf 12749 ioof 12836 fzf 12897 fzof 13036 axdc4uzlem 13352 sadcf 15802 smupf 15827 gcdf 15861 eucalgf 15927 vdwapf 16308 prdsval 16728 prdsplusg 16731 prdsmulr 16732 prdsvsca 16733 prdsds 16737 prdshom 16740 imasvscaf 16812 xpsff1o 16840 wunnat 17226 catcoppccl 17368 catcfuccl 17369 catcxpccl 17457 evlfcl 17472 hofcl 17509 mgmplusf 17862 grpsubf 18178 subgga 18430 lactghmga 18533 sylow1lem2 18724 sylow3lem1 18752 lsmssv 18768 smndlsmidm 18781 lsmidmOLD 18789 efgmf 18839 efgtf 18848 frgpuptf 18896 lmodscaf 19656 evlslem2 20292 xrsds 20588 phlipf 20796 mamucl 21010 matbas2d 21032 mamumat1cl 21048 ordtbas2 21799 iccordt 21822 txuni2 22173 xkotf 22193 txbasval 22214 tx1stc 22258 xkococn 22268 cnmpt12 22275 cnmpt21 22279 cnmpt2t 22281 cnmpt22 22282 cnmptcom 22286 cnmpt2k 22296 txswaphmeo 22413 xpstopnlem1 22417 cnmpt2plusg 22696 cnmpt2vsca 22803 prdsdsf 22977 blfvalps 22993 blfps 23016 blf 23017 stdbdmet 23126 met2ndci 23132 dscmet 23182 xrsxmet 23417 cnmpt2ds 23451 cnmpopc 23532 iimulcn 23542 ishtpy 23576 reparphti 23601 cnmpt2ip 23851 bcthlem5 23931 rrxmet 24011 dyadf 24192 itg1addlem2 24298 mbfi1fseqlem1 24316 mbfi1fseqlem3 24318 mbfi1fseqlem4 24319 mbfi1fseqlem5 24320 cxpcn3 25329 sgmf 25722 midf 26562 grpodivf 28315 nvmf 28422 ipf 28490 hvsubf 28792 ofoprabco 30409 suppovss 30426 fedgmullem1 31025 fedgmullem2 31026 fedgmul 31027 sitmf 31610 cvxsconn 32490 cvmlift2lem5 32554 uncf 34886 mblfinlem1 34944 mblfinlem2 34945 sdclem1 35033 metf1o 35045 rrnval 35120 rrnmet 35122 resubf 39260 frmx 39559 frmy 39560 icof 41531 |
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