Theorem feqresmpt 6962

Description: Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)

Hypotheses

Ref	Expression
feqmptd.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
feqresmpt.2	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
feqresmpt	⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

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

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem feqresmpt

Step	Hyp	Ref	Expression
1		feqmptd.1	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		feqresmpt.2	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
3	1, 2	fssresd 6759	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6961	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6911	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5252	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	eqtrdi 2787	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3949   ↦ cmpt 5232   ↾ cres 5679  ⟶wf 6540  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552

This theorem is referenced by:  pwfseqlem5  10661  pfxres  14634  gsumpt  19872  dpjidcl  19970  regsumsupp  21395  tsmsxplem2  23879  dvmulbr  25689  dvlip  25743  lhop1lem  25763  loglesqrt  26499  jensenlem1  26724  jensen  26726  amgm  26728  ushgredgedg  28750  ushgredgedgloop  28752  mgcf1o  32437  gsumzresunsn  32473  gsumpart  32474  gsumhashmul  32475  gsumle  32509  coinflippv  33777  fdvposlt  33906  fdvposle  33908  logdivsqrle  33957  gg-dvmulbr  35462  ftc1cnnclem  36863  dvasin  36876  dvacos  36877  dvreasin  36878  dvreacos  36879  areacirclem1  36880  dvrelog2  41236  dvrelog3  41237  cantnf2  42378  limsupvaluz2  44754  supcnvlimsup  44756  itgperiod  44997  fourierdlem69  45191  fourierdlem73  45195  fourierdlem74  45196  fourierdlem75  45197  fourierdlem76  45198  fourierdlem81  45203  fourierdlem85  45207  fourierdlem88  45210  fourierdlem92  45214  fourierdlem97  45219  fourierdlem100  45222  fourierdlem101  45223  fourierdlem103  45225  fourierdlem104  45226  fourierdlem107  45229  fourierdlem111  45233  fourierdlem112  45234  fouriersw  45247  sge0tsms  45396  sge0resrnlem  45419  meadjiunlem  45481  omeunle  45532  isomenndlem  45546