Theorem 2ndcof 6263

Description: Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
2ndcof	⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶)

Proof of Theorem 2ndcof

Step	Hyp	Ref	Expression
1		fo2nd 6257	. . . 4 ⊢ 2nd :V–onto→V
2		fofn 5512	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . 3 ⊢ 2nd Fn V
4		ffn 5435	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5		dffn2 5437	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
6	4, 5	sylib 122	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7		fnfco 5462	. . 3 ⊢ ((2nd Fn V ∧ 𝐹:𝐴⟶V) → (2nd ∘ 𝐹) Fn 𝐴)
8	3, 6, 7	sylancr 414	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹) Fn 𝐴)
9		rnco 5198	. . 3 ⊢ ran (2nd ∘ 𝐹) = ran (2nd ↾ ran 𝐹)
10		frn 5444	. . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11		ssres2 4995	. . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)))
12		rnss 4917	. . . . 5 ⊢ ((2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶)))
13	10, 11, 12	3syl 17	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶)))
14		f2ndres 6259	. . . . 5 ⊢ (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶
15		frn 5444	. . . . 5 ⊢ ((2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 → ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶)
16	14, 15	ax-mp 5	. . . 4 ⊢ ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶
17	13, 16	sstrdi 3209	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶)
18	9, 17	eqsstrid 3243	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ∘ 𝐹) ⊆ 𝐶)
19		df-f 5284	. 2 ⊢ ((2nd ∘ 𝐹):𝐴⟶𝐶 ↔ ((2nd ∘ 𝐹) Fn 𝐴 ∧ ran (2nd ∘ 𝐹) ⊆ 𝐶))
20	8, 18, 19	sylanbrc 417	1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶)

Syntax hints:   → wi 4  Vcvv 2773   ⊆ wss 3170   × cxp 4681  ran crn 4684   ↾ cres 4685   ∘ ccom 4687   Fn wfn 5275  ⟶wf 5276  –onto→wfo 5278  2^nd c2nd 6238

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fo 5286  df-fv 5288  df-2nd 6240

This theorem is referenced by: (None)