Theorem 2ndcof 6336

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 6330	. . . 4 ⊢ 2nd :V–onto→V
2		fofn 5570	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . 3 ⊢ 2nd Fn V
4		ffn 5489	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5		dffn2 5491	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
6	4, 5	sylib 122	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7		fnfco 5519	. . 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 5250	. . 3 ⊢ ran (2nd ∘ 𝐹) = ran (2nd ↾ ran 𝐹)
10		frn 5498	. . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11		ssres2 5046	. . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)))
12		rnss 4968	. . . . 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 6332	. . . . 5 ⊢ (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶
15		frn 5498	. . . . 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 3240	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶)
18	9, 17	eqsstrid 3274	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ∘ 𝐹) ⊆ 𝐶)
19		df-f 5337	. 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 2803   ⊆ wss 3201   × cxp 4729  ran crn 4732   ↾ cres 4733   ∘ ccom 4735   Fn wfn 5328  ⟶wf 5329  –onto→wfo 5331  2^nd c2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-2nd 6313

This theorem is referenced by: (None)