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| Mirrors > Home > MPE Home > Th. List > eqfnfv | Structured version Visualization version GIF version | ||
| Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
| Ref | Expression |
|---|---|
| eqfnfv | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dffn5 6967 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
| 2 | dffn5 6967 | . . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) | |
| 3 | eqeq12 2754 | . . 3 ⊢ ((𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ∧ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) → (𝐹 = 𝐺 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))) | |
| 4 | 1, 2, 3 | syl2anb 598 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))) |
| 5 | fvex 6919 | . . . 4 ⊢ (𝐹‘𝑥) ∈ V | |
| 6 | 5 | rgenw 3065 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V |
| 7 | mpteqb 7035 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
| 8 | 6, 7 | ax-mp 5 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
| 9 | 4, 8 | bitrdi 287 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ↦ cmpt 5225 Fn wfn 6556 ‘cfv 6561 |
| 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-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: eqfnfv2 7052 eqfnfvd 7054 eqfnfv2f 7055 eqfnun 7057 fvreseq0 7058 fnmptfvd 7061 fndmdifeq0 7064 fneqeql 7066 fnnfpeq0 7198 fprb 7214 fconst2g 7223 cocan1 7311 cocan2 7312 weniso 7374 fsplitfpar 8143 fnsuppres 8216 tfr3 8439 ixpfi2 9390 fipreima 9398 updjud 9974 fseqenlem1 10064 fpwwe2lem7 10677 ofsubeq0 12263 ser0f 14096 hashgval2 14417 hashf1lem1 14494 prodf1f 15928 efcvgfsum 16122 prmreclem2 16955 1arithlem4 16964 1arith 16965 smndex1n0mnd 18925 isgrpinv 19011 dprdf11 20043 frlmplusgvalb 21789 frlmvscavalb 21790 islindf4 21858 psrbagconf1o 21949 pthaus 23646 xkohaus 23661 cnmpt11 23671 cnmpt21 23679 prdsxmetlem 24378 rrxmet 25442 rolle 26028 tdeglem4 26099 resinf1o 26578 dchrelbas2 27281 dchreq 27302 eqeefv 28918 axlowdimlem14 28970 elntg2 29000 nmlno0lem 30812 phoeqi 30876 occllem 31322 dfiop2 31772 hoeq 31779 ho01i 31847 hoeq1 31849 kbpj 31975 nmlnop0iALT 32014 lnopco0i 32023 nlelchi 32080 rnbra 32126 kbass5 32139 hmopidmchi 32170 hmopidmpji 32171 pjssdif2i 32193 pjinvari 32210 bnj1542 34871 bnj580 34927 subfacp1lem3 35187 subfacp1lem5 35189 mrsubff1 35519 msubff1 35561 faclimlem1 35743 rdgprc 35795 broucube 37661 cocanfo 37726 sdclem2 37749 rrnmet 37836 rrnequiv 37842 ltrnid 40137 ltrneq2 40150 tendoeq1 40766 sticksstones1 42147 pw2f1ocnv 43049 caofcan 44342 addrcom 44494 fsneq 45211 dvnprodlem1 45961 cfsetsnfsetf1 47071 cfsetsnfsetfo 47072 rrx2pnecoorneor 48636 rrx2linest 48663 |
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