Theorem fnov 7574

Description: Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
fnov	⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem fnov

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5 6966	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)))
2		fveq2 6908	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7446	. . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2816	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝑥𝐹𝑦))
5	4	mpompt 7557	. . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))
6	5	eqeq2i 2776	. 2 ⊢ (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
7	1, 6	bitri 278	1 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ⟨cop 4838   ↦ cmpt 5189   × cxp 5653   Fn wfn 6553  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fn 6561  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448

This theorem is used by:  mapxpen  9190  dfioo2  13608  fnhomeqhomf  17894  reschomf  18035  cofulid  18094  cofurid  18095  prf1st  18407  prf2nd  18408  1st2ndprf  18409  curfuncf  18441  curf2ndf  18450  plusfeq  18853  scafeq  21187  cnfldadd  21714  cnfldmul  21716  cnfldsub  21736  ipfeq  21986  psrvscafval  22286  mdetunilem7  22963  madurid  22989  matunitlindflem1  23024  cnmpt22f  24024  cnmptcom  24027  xkocnv  24163  qustgplem  24470  stdbdxmet  24864  rrxds  25744  rrxmfval  25757  cnnvm  31314  ofpreima  33331  ressplusf  33594  elrgspnlem2  33874  fedgmullem2  34332  matmpo  34505  mndpluscn  34628  raddcn  34631  txsconnlem  36172  cvmlift2lem6  36240  cvmlift2lem7  36241  cvmlift2lem12  36246  unccur  38718  rngchomrnghmresALTV  49581  2arymaptfo  49971  isofval2  50351  funcf2lem2  50401  upeu4  50517  diag1  50631  fucofulem2  50638