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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 6967 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧))) | |
| 2 | fveq2 6906 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝐹‘〈𝑥, 𝑦〉)) | |
| 3 | df-ov 7434 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝑥𝐹𝑦)) |
| 5 | 4 | mpompt 7547 | . . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)) |
| 6 | 5 | eqeq2i 2750 | . 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 206 = wceq 1540 〈cop 4632 ↦ cmpt 5225 × cxp 5683 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 |
| 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-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-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 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-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: mapxpen 9183 dfioo2 13490 fnhomeqhomf 17734 reschomf 17875 cofulid 17935 cofurid 17936 prf1st 18249 prf2nd 18250 1st2ndprf 18251 curfuncf 18283 curf2ndf 18292 plusfeq 18661 scafeq 20880 cnfldadd 21370 cnfldmul 21372 dfcnfldOLD 21380 cnfldsub 21410 ipfeq 21668 psrvscafval 21968 mdetunilem7 22624 madurid 22650 cnmpt22f 23683 cnmptcom 23686 xkocnv 23822 qustgplem 24129 stdbdxmet 24528 iimulcnOLD 24968 rrxds 25427 rrxmfval 25440 cnnvm 30701 ofpreima 32675 ressplusf 32948 elrgspnlem2 33247 fedgmullem2 33681 matmpo 33802 mndpluscn 33925 raddcn 33928 txsconnlem 35245 cvmlift2lem6 35313 cvmlift2lem7 35314 cvmlift2lem12 35319 unccur 37610 matunitlindflem1 37623 rngchomrnghmresALTV 48195 2arymaptfo 48575 funcf2lem2 48915 upeu4 48947 diag1 49004 fucofulem2 49006 |
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