Theorem cores 5134

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 2742	. . . . . . 7 ⊢ 𝑧 ∈ V
2		vex 2742	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 4862	. . . . . 6 ⊢ (𝑧𝐵𝑦 → 𝑦 ∈ ran 𝐵)
4		ssel 3151	. . . . . 6 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ 𝐶))
5		vex 2742	. . . . . . . 8 ⊢ 𝑥 ∈ V
6	5	brres 4915	. . . . . . 7 ⊢ (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 ∈ 𝐶))
7	6	rbaib 921	. . . . . 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 450	. . . 4 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)))
10	9	exbidv 1825	. . 3 ⊢ (ran 𝐵 ⊆ 𝐶 → (∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)))
11	10	opabbidv 4071	. 2 ⊢ (ran 𝐵 ⊆ 𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)})
12		df-co 4637	. 2 ⊢ ((𝐴 ↾ 𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)}
13		df-co 4637	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)}
14	11, 12, 13	3eqtr4g 2235	1 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148   ⊆ wss 3131   class class class wbr 4005  {copab 4065  ran crn 4629   ↾ cres 4630   ∘ ccom 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640

This theorem is referenced by:  cocnvcnv1  5141  cores2  5143  cocnvres  5155  relcoi2  5161  fco2  5384  fcoi2  5399