Theorem eqfnfv 6267

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 6198	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2		dffn5 6198	. . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
3		eqeq12 2634	. . 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 496	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))))
5		fvex 6158	. . . 4 ⊢ (𝐹‘𝑥) ∈ V
6	5	rgenw 2919	. . 3 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V
7		mpteqb 6255	. . 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 276	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   ↦ cmpt 4673   Fn wfn 5842  ‘cfv 5847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855

This theorem is referenced by:  eqfnfv2  6268  eqfnfvd  6270  eqfnfv2f  6271  fvreseq0  6273  fnmptfvd  6276  fndmdifeq0  6279  fneqeql  6281  fnnfpeq0  6398  fconst2g  6422  cocan1  6500  cocan2  6501  weniso  6558  fnsuppres  7267  tfr3  7440  ixpfi2  8208  fipreima  8216  fseqenlem1  8791  fpwwe2lem8  9403  ofsubeq0  10961  ser0f  12794  hashgval2  13107  hashf1lem1  13177  prodf1f  14549  efcvgfsum  14741  prmreclem2  15545  1arithlem4  15554  1arith  15555  isgrpinv  17393  dprdf11  18343  psrbagconf1o  19293  islindf4  20096  pthaus  21351  xkohaus  21366  cnmpt11  21376  cnmpt21  21384  prdsxmetlem  22083  rrxmet  23099  rolle  23657  tdeglem4  23724  resinf1o  24186  dchrelbas2  24862  dchreq  24883  eqeefv  25683  axlowdimlem14  25735  nmlno0lem  27494  phoeqi  27559  occllem  28008  dfiop2  28458  hoeq  28465  ho01i  28533  hoeq1  28535  kbpj  28661  nmlnop0iALT  28700  lnopco0i  28709  nlelchi  28766  rnbra  28812  kbass5  28825  hmopidmchi  28856  hmopidmpji  28857  pjssdif2i  28879  pjinvari  28896  bnj1542  30632  bnj580  30688  subfacp1lem3  30869  subfacp1lem5  30871  mrsubff1  31116  msubff1  31158  faclimlem1  31334  fprb  31370  rdgprc  31398  broucube  33072  cocanfo  33141  eqfnun  33145  sdclem2  33167  rrnmet  33257  rrnequiv  33263  ltrnid  34898  ltrneq2  34911  tendoeq1  35529  pw2f1ocnv  37081  caofcan  38001  addrcom  38158  fsneq  38869  dvnprodlem1  39464