Theorem cores 5050

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 2692	. . . . . . 7 ⊢ 𝑧 ∈ V
2		vex 2692	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 4780	. . . . . 6 ⊢ (𝑧𝐵𝑦 → 𝑦 ∈ ran 𝐵)
4		ssel 3096	. . . . . 6 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ 𝐶))
5		vex 2692	. . . . . . . 8 ⊢ 𝑥 ∈ V
6	5	brres 4833	. . . . . . 7 ⊢ (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 ∈ 𝐶))
7	6	rbaib 907	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥))
8	3, 4, 7	syl56 34	. . . . 5 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥)))
9	8	pm5.32d 446	. . . 4 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)))
10	9	exbidv 1798	. . 3 ⊢ (ran 𝐵 ⊆ 𝐶 → (∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)))
11	10	opabbidv 4002	. 2 ⊢ (ran 𝐵 ⊆ 𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)})
12		df-co 4556	. 2 ⊢ ((𝐴 ↾ 𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)}
13		df-co 4556	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)}
14	11, 12, 13	3eqtr4g 2198	1 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481   ⊆ wss 3076   class class class wbr 3937  {copab 3996  ran crn 4548   ↾ cres 4549   ∘ ccom 4551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559

This theorem is referenced by:  cocnvcnv1  5057  cores2  5059  cocnvres  5071  relcoi2  5077  fco2  5297  fcoi2  5312