Theorem eqfnfv 6560

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.)

Assertion

Ref	Expression
eqfnfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺

Proof of Theorem eqfnfv

Step	Hyp	Ref	Expression
1		dffn5 6488	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2		dffn5 6488	. . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
3		eqeq12 2838	. . 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 591	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))))
5		fvex 6446	. . . 4 ⊢ (𝐹‘𝑥) ∈ V
6	5	rgenw 3133	. . 3 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V
7		mpteqb 6546	. . 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 279	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414   ↦ cmpt 4952   Fn wfn 6118  ‘cfv 6123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131

This theorem is referenced by:  eqfnfv2  6561  eqfnfvd  6563  eqfnfv2f  6564  fvreseq0  6566  fnmptfvd  6569  fndmdifeq0  6572  fneqeql  6574  fnnfpeq0  6696  fconst2g  6724  cocan1  6801  cocan2  6802  weniso  6859  fnsuppres  7587  tfr3  7761  ixpfi2  8533  fipreima  8541  updjud  9073  fseqenlem1  9160  fpwwe2lem8  9774  ofsubeq0  11347  ser0f  13148  hashgval2  13457  hashf1lem1  13528  prodf1f  14997  efcvgfsum  15188  prmreclem2  15992  1arithlem4  16001  1arith  16002  isgrpinv  17826  dprdf11  18776  psrbagconf1o  19735  frlmplusgvalb  20475  frlmvscavalb  20476  islindf4  20544  pthaus  21812  xkohaus  21827  cnmpt11  21837  cnmpt21  21845  prdsxmetlem  22543  rrxmet  23576  rolle  24152  tdeglem4  24219  resinf1o  24682  dchrelbas2  25375  dchreq  25396  eqeefv  26202  axlowdimlem14  26254  elntg2  26284  nmlno0lem  28192  phoeqi  28257  occllem  28706  dfiop2  29156  hoeq  29163  ho01i  29231  hoeq1  29233  kbpj  29359  nmlnop0iALT  29398  lnopco0i  29407  nlelchi  29464  rnbra  29510  kbass5  29523  hmopidmchi  29554  hmopidmpji  29555  pjssdif2i  29577  pjinvari  29594  bnj1542  31462  bnj580  31518  subfacp1lem3  31699  subfacp1lem5  31701  mrsubff1  31946  msubff1  31988  faclimlem1  32160  fprb  32200  rdgprc  32227  broucube  33980  cocanfo  34048  eqfnun  34052  sdclem2  34073  rrnmet  34163  rrnequiv  34169  ltrnid  36203  ltrneq2  36216  tendoeq1  36832  pw2f1ocnv  38440  caofcan  39355  addrcom  39510  fsneq  40197  dvnprodlem1  40949  rrx2linest  43289