Theorem feqresmpt 6392

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 6211	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6391	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6348	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 4874	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	syl6eq 2821	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1631   ⊆ wss 3723   ↦ cmpt 4863   ↾ cres 5251  ⟶wf 6027  ‘cfv 6031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039

This theorem is referenced by:  pwfseqlem5  9687  swrd0val  13629  gsumpt  18568  dpjidcl  18665  regsumsupp  20185  tsmsxplem2  22177  dvmulbr  23922  dvlip  23976  lhop1lem  23996  loglesqrt  24720  jensenlem1  24934  jensen  24936  amgm  24938  ushgredgedg  26343  ushgredgedgloop  26345  ushgredgedgloopOLD  26346  gsumle  30119  coinflippv  30885  fdvposlt  31017  fdvposle  31019  logdivsqrle  31068  ftc1cnnclem  33815  dvasin  33828  dvacos  33829  dvreasin  33830  dvreacos  33831  areacirclem1  33832  limsupvaluz2  40488  supcnvlimsup  40490  itgperiod  40714  fourierdlem69  40909  fourierdlem73  40913  fourierdlem74  40914  fourierdlem75  40915  fourierdlem76  40916  fourierdlem81  40921  fourierdlem85  40925  fourierdlem88  40928  fourierdlem92  40932  fourierdlem97  40937  fourierdlem100  40940  fourierdlem101  40941  fourierdlem103  40943  fourierdlem104  40944  fourierdlem107  40947  fourierdlem111  40951  fourierdlem112  40952  fouriersw  40965  sge0tsms  41114  sge0resrnlem  41137  meadjiunlem  41199  omeunle  41250  isomenndlem  41264  pfxres  41916