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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 6961 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧))) | |
2 | fveq2 6901 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | df-ov 7427 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | 2, 3 | eqtr4di 2784 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝑥𝐹𝑦)) |
5 | 4 | mpompt 7539 | . . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)) |
6 | 5 | eqeq2i 2739 | . 2 ⊢ (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
7 | 1, 6 | bitri 274 | 1 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 〈cop 4639 ↦ cmpt 5236 × cxp 5680 Fn wfn 6549 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fn 6557 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 |
This theorem is referenced by: mapxpen 9181 dfioo2 13481 fnhomeqhomf 17704 reschomf 17848 cofulid 17909 cofurid 17910 prf1st 18228 prf2nd 18229 1st2ndprf 18230 curfuncf 18263 curf2ndf 18272 plusfeq 18641 scafeq 20858 cnfldadd 21349 cnfldmul 21351 dfcnfldOLD 21359 cnfldsub 21389 ipfeq 21646 psrvscafval 21957 mdetunilem7 22611 madurid 22637 cnmpt22f 23670 cnmptcom 23673 xkocnv 23809 qustgplem 24116 stdbdxmet 24515 iimulcnOLD 24953 rrxds 25412 rrxmfval 25425 cnnvm 30615 ofpreima 32582 ressplusf 32827 fedgmullem2 33525 matmpo 33618 mndpluscn 33741 raddcn 33744 txsconnlem 35068 cvmlift2lem6 35136 cvmlift2lem7 35137 cvmlift2lem12 35142 unccur 37304 matunitlindflem1 37317 rngchomrnghmresALTV 47656 2arymaptfo 48042 |
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