Theorem feqresmpt 6602

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 6413	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
4	3	feqmptd 6601	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
5		fvres 6557	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
6	5	mpteq2ia 5051	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
7	4, 6	syl6eq 2847	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1522   ⊆ wss 3859   ↦ cmpt 5041   ↾ cres 5445  ⟶wf 6221  ‘cfv 6225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233

This theorem is referenced by:  pwfseqlem5  9931  pfxres  13877  gsumpt  18802  dpjidcl  18897  regsumsupp  20448  tsmsxplem2  22445  dvmulbr  24219  dvlip  24273  lhop1lem  24293  loglesqrt  25020  jensenlem1  25246  jensen  25248  amgm  25250  ushgredgedg  26694  ushgredgedgloop  26696  gsumle  30494  gsumzresunsn  30502  coinflippv  31358  fdvposlt  31487  fdvposle  31489  logdivsqrle  31538  ftc1cnnclem  34496  dvasin  34509  dvacos  34510  dvreasin  34511  dvreacos  34512  areacirclem1  34513  limsupvaluz2  41561  supcnvlimsup  41563  itgperiod  41807  fourierdlem69  42002  fourierdlem73  42006  fourierdlem74  42007  fourierdlem75  42008  fourierdlem76  42009  fourierdlem81  42014  fourierdlem85  42018  fourierdlem88  42021  fourierdlem92  42025  fourierdlem97  42030  fourierdlem100  42033  fourierdlem101  42034  fourierdlem103  42036  fourierdlem104  42037  fourierdlem107  42040  fourierdlem111  42044  fourierdlem112  42045  fouriersw  42058  sge0tsms  42204  sge0resrnlem  42227  meadjiunlem  42289  omeunle  42340  isomenndlem  42354