Theorem 2ndcof 6143

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 6137	. . . 4 ⊢ 2nd :V–onto→V
2		fofn 5422	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . 3 ⊢ 2nd Fn V
4		ffn 5347	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5		dffn2 5349	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
6	4, 5	sylib 121	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7		fnfco 5372	. . 3 ⊢ ((2nd Fn V ∧ 𝐹:𝐴⟶V) → (2nd ∘ 𝐹) Fn 𝐴)
8	3, 6, 7	sylancr 412	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹) Fn 𝐴)
9		rnco 5117	. . 3 ⊢ ran (2nd ∘ 𝐹) = ran (2nd ↾ ran 𝐹)
10		frn 5356	. . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11		ssres2 4918	. . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)))
12		rnss 4841	. . . . 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 6139	. . . . 5 ⊢ (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶
15		frn 5356	. . . . 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 3159	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶)
18	9, 17	eqsstrid 3193	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ∘ 𝐹) ⊆ 𝐶)
19		df-f 5202	. 2 ⊢ ((2nd ∘ 𝐹):𝐴⟶𝐶 ↔ ((2nd ∘ 𝐹) Fn 𝐴 ∧ ran (2nd ∘ 𝐹) ⊆ 𝐶))
20	8, 18, 19	sylanbrc 415	1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶)

Syntax hints:   → wi 4  Vcvv 2730   ⊆ wss 3121   × cxp 4609  ran crn 4612   ↾ cres 4613   ∘ ccom 4615   Fn wfn 5193  ⟶wf 5194  –onto→wfo 5196  2^nd c2nd 6118

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-2nd 6120

This theorem is referenced by: (None)