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Theorem 1stcof 8002

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 7992	. . . 4 ⊢ 1^st :V–onto→V
2		fofn 6805	. . . 4 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . 3 ⊢ 1^st Fn V
4		ffn 6715	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5		dffn2 6717	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
6	4, 5	sylib 217	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7		fnfco 6754	. . 3 ⊢ ((1^st Fn V ∧ 𝐹:𝐴⟶V) → (1^st ∘ 𝐹) Fn 𝐴)
8	3, 6, 7	sylancr 588	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹) Fn 𝐴)
9		rnco 6249	. . 3 ⊢ ran (1^st ∘ 𝐹) = ran (1^st ↾ ran 𝐹)
10		frn 6722	. . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11		ssres2 6008	. . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1^st ↾ ran 𝐹) ⊆ (1^st ↾ (𝐵 × 𝐶)))
12		rnss 5937	. . . . 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 7996	. . . . 5 ⊢ (1^st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15		frn 6722	. . . . 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	sstrdi 3994	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ↾ ran 𝐹) ⊆ 𝐵)
18	9, 17	eqsstrid 4030	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ∘ 𝐹) ⊆ 𝐵)
19		df-f 6545	. 2 ⊢ ((1^st ∘ 𝐹):𝐴⟶𝐵 ↔ ((1^st ∘ 𝐹) Fn 𝐴 ∧ ran (1^st ∘ 𝐹) ⊆ 𝐵))
20	8, 18, 19	sylanbrc 584	1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹):𝐴⟶𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 Vcvv 3475 ⊆ wss 3948 × cxp 5674 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 Fn wfn 6536 ⟶wf 6537 –onto→wfo 6539 1^st c1st 7970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6543 df-fn 6544 df-f 6545 df-fo 6547 df-1st 7972

This theorem is referenced by: ruclem11 16180 ruclem12 16181 caubl 24817

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