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Theorem 1stcof 7359

Description: Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)

**Assertion**
Ref	Expression
1stcof	⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹):𝐴⟶𝐵)

**Proof of Theorem 1stcof**
Step	Hyp	Ref	Expression
1		fo1st 7349	. . . 4 ⊢ 1^st :V–onto→V
2		fofn 6274	. . . 4 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . 3 ⊢ 1^st Fn V
4		ffn 6202	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5		dffn2 6204	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
6	4, 5	sylib 208	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7		fnfco 6226	. . 3 ⊢ ((1^st Fn V ∧ 𝐹:𝐴⟶V) → (1^st ∘ 𝐹) Fn 𝐴)
8	3, 6, 7	sylancr 698	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹) Fn 𝐴)
9		rnco 5798	. . 3 ⊢ ran (1^st ∘ 𝐹) = ran (1^st ↾ ran 𝐹)
10		frn 6210	. . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11		ssres2 5579	. . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1^st ↾ ran 𝐹) ⊆ (1^st ↾ (𝐵 × 𝐶)))
12		rnss 5505	. . . . 5 ⊢ ((1^st ↾ ran 𝐹) ⊆ (1^st ↾ (𝐵 × 𝐶)) → ran (1^st ↾ ran 𝐹) ⊆ ran (1^st ↾ (𝐵 × 𝐶)))
13	10, 11, 12	3syl 18	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ↾ ran 𝐹) ⊆ ran (1^st ↾ (𝐵 × 𝐶)))
14		f1stres 7353	. . . . 5 ⊢ (1^st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15		frn 6210	. . . . 5 ⊢ ((1^st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 → ran (1^st ↾ (𝐵 × 𝐶)) ⊆ 𝐵)
16	14, 15	ax-mp 5	. . . 4 ⊢ ran (1^st ↾ (𝐵 × 𝐶)) ⊆ 𝐵
17	13, 16	syl6ss 3752	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ↾ ran 𝐹) ⊆ 𝐵)
18	9, 17	syl5eqss 3786	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ∘ 𝐹) ⊆ 𝐵)
19		df-f 6049	. 2 ⊢ ((1^st ∘ 𝐹):𝐴⟶𝐵 ↔ ((1^st ∘ 𝐹) Fn 𝐴 ∧ ran (1^st ∘ 𝐹) ⊆ 𝐵))
20	8, 18, 19	sylanbrc 701	1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹):𝐴⟶𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 Vcvv 3336 ⊆ wss 3711 × cxp 5260 ran crn 5263 ↾ cres 5264 ∘ ccom 5266 Fn wfn 6040 ⟶wf 6041 –onto→wfo 6043 1^st c1st 7327

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-sep 4929 ax-nul 4937 ax-pr 5051 ax-un 7110

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-ral 3051 df-rex 3052 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-nul 4055 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4585 df-iun 4670 df-br 4801 df-opab 4861 df-mpt 4878 df-id 5170 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-fo 6051 df-fv 6053 df-1st 7329

This theorem is referenced by: ruclem11 15164 ruclem12 15165 caubl 23302

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