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Theorem 1stcof 8052

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 8042	. . . 4 ⊢ 1^st :V⟶ontoV
2		fofn 6817	. . . 4 ⊢ (1^st :V⟶ontoV → 1^st Fn V)
3	1, 2	ax-mp 5	. . 3 ⊢ 1^st Fn V
4		ffn 6728	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5		dffn2 6730	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
6	4, 5	sylib 221	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7		fnfco 6766	. . 3 ⊢ ((1^st Fn V ∧ 𝐹:𝐴⟶V) → (1^st ∘ 𝐹) Fn 𝐴)
8	3, 6, 7	sylancr 599	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹) Fn 𝐴)
9		rnco 6265	. . 3 ⊢ ran (1^st ∘ 𝐹) = ran (1^st ↾ ran 𝐹)
10		frn 6736	. . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11		ssres2 6005	. . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1^st ↾ ran 𝐹) ⊆ (1^st ↾ (𝐵 × 𝐶)))
12		rnss 5927	. . . . 5 ⊢ ((1^st ↾ ran 𝐹) ⊆ (1^st ↾ (𝐵 × 𝐶)) → ran (1^st ↾ ran 𝐹) ⊆ ran (1^st ↾ (𝐵 × 𝐶)))
13	10, 11, 12	3syl 19	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ↾ ran 𝐹) ⊆ ran (1^st ↾ (𝐵 × 𝐶)))
14		f1stres 8046	. . . . 5 ⊢ (1^st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15		frn 6736	. . . . 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 3925	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ↾ ran 𝐹) ⊆ 𝐵)
18	9, 17	eqsstrid 3951	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ∘ 𝐹) ⊆ 𝐵)
19		df-f 6559	. 2 ⊢ ((1^st ∘ 𝐹):𝐴⟶𝐵 ↔ ((1^st ∘ 𝐹) Fn 𝐴 ∧ ran (1^st ∘ 𝐹) ⊆ 𝐵))
20	8, 18, 19	sylanbrc 595	1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹):𝐴⟶𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 Vcvv 3454 ⊆ wss 3900 × cxp 5653 ran crn 5656 ↾ cres 5657 ∘ ccom 5659 Fn wfn 6550 ⟶wf 6551 ⟶ontowfo 6553 1^st c1st 8020

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-pr 5395 ax-un 7772

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-fun 6557 df-fn 6558 df-f 6559 df-fo 6561 df-1st 8022

This theorem is used by: ruclem11 16434 ruclem12 16435 caubl 25656

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