Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6151 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
2 | dffn5 6151 | . . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) | |
3 | eqeq12 2623 | . . 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 495 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))) |
5 | fvex 6113 | . . . 4 ⊢ (𝐹‘𝑥) ∈ V | |
6 | 5 | rgenw 2908 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V |
7 | mpteqb 6207 | . . 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 | syl6bb 275 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ↦ cmpt 4643 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: eqfnfv2 6220 eqfnfvd 6222 eqfnfv2f 6223 fvreseq0 6225 fnmptfvd 6228 fndmdifeq0 6231 fneqeql 6233 fnnfpeq0 6349 fconst2g 6373 cocan1 6446 cocan2 6447 weniso 6504 fnsuppres 7209 tfr3 7382 ixpfi2 8147 fipreima 8155 fseqenlem1 8730 fpwwe2lem8 9338 ofsubeq0 10894 ser0f 12716 hashgval2 13028 hashf1lem1 13096 prodf1f 14463 efcvgfsum 14655 prmreclem2 15459 1arithlem4 15468 1arith 15469 isgrpinv 17295 dprdf11 18245 psrbagconf1o 19195 islindf4 19996 pthaus 21251 xkohaus 21266 cnmpt11 21276 cnmpt21 21284 prdsxmetlem 21983 rrxmet 22999 rolle 23557 tdeglem4 23624 resinf1o 24086 dchrelbas2 24762 dchreq 24783 eqeefv 25583 axlowdimlem14 25635 nmlno0lem 27032 phoeqi 27097 occllem 27546 dfiop2 27996 hoeq 28003 ho01i 28071 hoeq1 28073 kbpj 28199 nmlnop0iALT 28238 lnopco0i 28247 nlelchi 28304 rnbra 28350 kbass5 28363 hmopidmchi 28394 hmopidmpji 28395 pjssdif2i 28417 pjinvari 28434 bnj1542 30181 bnj580 30237 subfacp1lem3 30418 subfacp1lem5 30420 mrsubff1 30665 msubff1 30707 faclimlem1 30882 fprb 30916 rdgprc 30944 broucube 32613 cocanfo 32682 eqfnun 32686 sdclem2 32708 rrnmet 32798 rrnequiv 32804 ltrnid 34439 ltrneq2 34452 tendoeq1 35070 pw2f1ocnv 36622 caofcan 37544 addrcom 37700 fsneq 38393 dvnprodlem1 38836 |
Copyright terms: Public domain | W3C validator |