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Mirrors > Home > ILE Home > Th. List > feqmptd | GIF version |
Description: Deduction form of dffn5im 5435. (Contributed by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
feqmptd.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
feqmptd | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feqmptd.1 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | ffn 5242 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
4 | dffn5im 5435 | . 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
5 | 3, 4 | syl 14 | 1 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ↦ cmpt 3959 Fn wfn 5088 ⟶wf 5089 ‘cfv 5093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 |
This theorem is referenced by: feqresmpt 5443 cofmpt 5557 fcoconst 5559 suppssof1 5967 ofco 5968 caofinvl 5972 caofcom 5973 mapxpen 6710 xpmapenlem 6711 cnrecnv 10650 lmcn2 12376 cnmpt11f 12380 cnmpt21f 12388 cncfmpt1f 12680 negfcncf 12685 cnrehmeocntop 12689 dvcnp2cntop 12759 dvimulf 12766 dvcoapbr 12767 dvcj 12769 dvfre 12770 dvef 12783 |
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