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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 6717 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧))) | |
2 | fveq2 6663 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘𝑧) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | df-ov 7148 | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | 2, 3 | syl6eqr 2871 | . . . 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 2831 | . 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 4563 ↦ cmpt 5137 × cxp 5546 Fn wfn 6343 ‘cfv 6348 (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 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 |
This theorem is referenced by: mapxpen 8671 dfioo2 12826 fnhomeqhomf 16949 reschomf 17089 cofulid 17148 cofurid 17149 prf1st 17442 prf2nd 17443 1st2ndprf 17444 curfuncf 17476 curf2ndf 17485 plusfeq 17848 scafeq 19583 psrvscafval 20098 cnfldsub 20501 ipfeq 20722 mdetunilem7 21155 madurid 21181 cnmpt22f 22211 cnmptcom 22214 xkocnv 22350 qustgplem 22656 stdbdxmet 23052 iimulcn 23469 rrxds 23923 rrxmfval 23936 cnnvm 28386 ofpreima 30338 ressplusf 30564 fedgmullem2 30925 matmpo 30967 mndpluscn 31068 rmulccn 31070 raddcn 31071 txsconnlem 32384 cvmlift2lem6 32452 cvmlift2lem7 32453 cvmlift2lem12 32458 unccur 34756 matunitlindflem1 34769 rngchomrnghmresALTV 44195 |
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