Theorem cores 6204

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 3442	. . . . . . 7 ⊢ 𝑧 ∈ V
2		vex 3442	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 5889	. . . . . 6 ⊢ (𝑧𝐵𝑦 → 𝑦 ∈ ran 𝐵)
4		ssel 3925	. . . . . 6 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ 𝐶))
5		vex 3442	. . . . . . . 8 ⊢ 𝑥 ∈ V
6	5	brresi 5944	. . . . . . 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 1922	. . 3 ⊢ (ran 𝐵 ⊆ 𝐶 → (∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)))
11	10	opabbidv 5161	. 2 ⊢ (ran 𝐵 ⊆ 𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)})
12		df-co 5630	. 2 ⊢ ((𝐴 ↾ 𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)}
13		df-co 5630	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)}
14	11, 12, 13	3eqtr4g 2793	1 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ⊆ wss 3899   class class class wbr 5095  {copab 5157  ran crn 5622   ↾ cres 5623   ∘ ccom 5625

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633

This theorem is referenced by:  cocnvcnv1  6213  cores2  6215  relcoi2  6232  funresfunco  6530  fco2  6685  fcoi2  6706  f1ocoima  7246  domss2  9059  cottrcl  9619  canthp1lem2  10554  imasdsval2  17430  frmdss2  18781  gsumval3lem1  19827  gsumzres  19831  gsumzaddlem  19843  dprdf1  19957  kgencn2  23482  tsmsf1o  24070  lgamcvg2  27002  hhssims  31265  ccatws1f1olast  32944  symgcom  33063  cycpmconjslem1  33134  cycpmconjslem2  33135  eulerpartgbij  34396  cvmlift2lem9a  35358  poimirlem9  37679  fourierdlem53  46271  tposres3  48995