![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fnov | Structured version Visualization version GIF version |
Description: Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fnov | ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn5 6902 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧))) | |
2 | fveq2 6843 | . . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩)) | |
3 | df-ov 7361 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩) | |
4 | 2, 3 | eqtr4di 2791 | . . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝑥𝐹𝑦)) |
5 | 4 | mpompt 7471 | . . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)) |
6 | 5 | eqeq2i 2746 | . 2 ⊢ (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
7 | 1, 6 | bitri 275 | 1 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ⟨cop 4593 ↦ cmpt 5189 × cxp 5632 Fn wfn 6492 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 |
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-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 |
This theorem is referenced by: mapxpen 9090 dfioo2 13373 fnhomeqhomf 17576 reschomf 17720 cofulid 17781 cofurid 17782 prf1st 18097 prf2nd 18098 1st2ndprf 18099 curfuncf 18132 curf2ndf 18141 plusfeq 18510 scafeq 20357 cnfldsub 20841 ipfeq 21070 psrvscafval 21374 mdetunilem7 21983 madurid 22009 cnmpt22f 23042 cnmptcom 23045 xkocnv 23181 qustgplem 23488 stdbdxmet 23887 iimulcn 24317 rrxds 24773 rrxmfval 24786 cnnvm 29666 ofpreima 31627 ressplusf 31866 fedgmullem2 32382 matmpo 32441 mndpluscn 32564 rmulccn 32566 raddcn 32567 txsconnlem 33891 cvmlift2lem6 33959 cvmlift2lem7 33960 cvmlift2lem12 33965 unccur 36107 matunitlindflem1 36120 rngchomrnghmresALTV 46380 2arymaptfo 46826 |
Copyright terms: Public domain | W3C validator |