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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 6718 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧))) | |
2 | fveq2 6664 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | df-ov 7148 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | 2, 3 | syl6eqr 2874 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝑥𝐹𝑦)) |
5 | 4 | mpompt 7255 | . . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)) |
6 | 5 | eqeq2i 2834 | . 2 ⊢ (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
7 | 1, 6 | bitri 276 | 1 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 〈cop 4565 ↦ cmpt 5138 × cxp 5547 Fn wfn 6344 ‘cfv 6349 (class class class)co 7145 ∈ cmpo 7147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 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-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 |
This theorem is referenced by: mapxpen 8672 dfioo2 12828 fnhomeqhomf 16951 reschomf 17091 cofulid 17150 cofurid 17151 prf1st 17444 prf2nd 17445 1st2ndprf 17446 curfuncf 17478 curf2ndf 17487 plusfeq 17850 scafeq 19585 psrvscafval 20100 cnfldsub 20503 ipfeq 20724 mdetunilem7 21157 madurid 21183 cnmpt22f 22213 cnmptcom 22216 xkocnv 22352 qustgplem 22658 stdbdxmet 23054 iimulcn 23471 rrxds 23925 rrxmfval 23938 cnnvm 28387 ofpreima 30339 ressplusf 30565 fedgmullem2 30926 matmpo 30968 mndpluscn 31069 rmulccn 31071 raddcn 31072 txsconnlem 32385 cvmlift2lem6 32453 cvmlift2lem7 32454 cvmlift2lem12 32459 unccur 34757 matunitlindflem1 34770 rngchomrnghmresALTV 44165 |
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