Theorem cores 6225

Description: Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cores	⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵))

Proof of Theorem cores

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3454	. . . . . . 7 ⊢ 𝑧 ∈ V
2		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 5909	. . . . . 6 ⊢ (𝑧𝐵𝑦 → 𝑦 ∈ ran 𝐵)
4		ssel 3943	. . . . . 6 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ 𝐶))
5		vex 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
6	5	brresi 5962	. . . . . . 7 ⊢ (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦𝐴𝑥))
7	6	baib 535	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥))
8	3, 4, 7	syl56 36	. . . . 5 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥)))
9	8	pm5.32d 577	. . . 4 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)))
10	9	exbidv 1921	. . 3 ⊢ (ran 𝐵 ⊆ 𝐶 → (∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)))
11	10	opabbidv 5176	. 2 ⊢ (ran 𝐵 ⊆ 𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)})
12		df-co 5650	. 2 ⊢ ((𝐴 ↾ 𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)}
13		df-co 5650	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)}
14	11, 12, 13	3eqtr4g 2790	1 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ⊆ wss 3917   class class class wbr 5110  {copab 5172  ran crn 5642   ↾ cres 5643   ∘ ccom 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653

This theorem is referenced by:  cocnvcnv1  6233  cores2  6235  relcoi2  6253  funresfunco  6560  fco2  6717  fcoi2  6738  f1ocoima  7281  domss2  9106  cottrcl  9679  canthp1lem2  10613  imasdsval2  17486  frmdss2  18797  gsumval3lem1  19842  gsumzres  19846  gsumzaddlem  19858  dprdf1  19972  kgencn2  23451  tsmsf1o  24039  lgamcvg2  26972  hhssims  31210  ccatws1f1olast  32881  symgcom  33047  cycpmconjslem1  33118  cycpmconjslem2  33119  eulerpartgbij  34370  cvmlift2lem9a  35297  poimirlem9  37630  fourierdlem53  46164  tposres3  48873