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Theorem 1stcof 7861

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 7851	. . . 4 ⊢ 1^st :V–onto→V
2		fofn 6690	. . . 4 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . 3 ⊢ 1^st Fn V
4		ffn 6600	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5		dffn2 6602	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
6	4, 5	sylib 217	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7		fnfco 6639	. . 3 ⊢ ((1^st Fn V ∧ 𝐹:𝐴⟶V) → (1^st ∘ 𝐹) Fn 𝐴)
8	3, 6, 7	sylancr 587	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹) Fn 𝐴)
9		rnco 6156	. . 3 ⊢ ran (1^st ∘ 𝐹) = ran (1^st ↾ ran 𝐹)
10		frn 6607	. . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11		ssres2 5919	. . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1^st ↾ ran 𝐹) ⊆ (1^st ↾ (𝐵 × 𝐶)))
12		rnss 5848	. . . . 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 7855	. . . . 5 ⊢ (1^st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15		frn 6607	. . . . 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 3933	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ↾ ran 𝐹) ⊆ 𝐵)
18	9, 17	eqsstrid 3969	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ∘ 𝐹) ⊆ 𝐵)
19		df-f 6437	. 2 ⊢ ((1^st ∘ 𝐹):𝐴⟶𝐵 ↔ ((1^st ∘ 𝐹) Fn 𝐴 ∧ ran (1^st ∘ 𝐹) ⊆ 𝐵))
20	8, 18, 19	sylanbrc 583	1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹):𝐴⟶𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 Vcvv 3432 ⊆ wss 3887 × cxp 5587 ran crn 5590 ↾ cres 5591 ∘ ccom 5593 Fn wfn 6428 ⟶wf 6429 –onto→wfo 6431 1^st c1st 7829

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-fun 6435 df-fn 6436 df-f 6437 df-fo 6439 df-1st 7831

This theorem is referenced by: ruclem11 15949 ruclem12 15950 caubl 24472

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