1stcof - Intuitionistic Logic Explorer

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Theorem 1stcof 6370

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 6364	. . . 4 ⊢ 1^st :V–onto→V
2		fofn 5597	. . . 4 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . 3 ⊢ 1^st Fn V
4		ffn 5513	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴)
5		dffn2 5515	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
6	4, 5	sylib 122	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V)
7		fnfco 5544	. . 3 ⊢ ((1^st Fn V ∧ 𝐹:𝐴⟶V) → (1^st ∘ 𝐹) Fn 𝐴)
8	3, 6, 7	sylancr 414	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹) Fn 𝐴)
9		rnco 5274	. . 3 ⊢ ran (1^st ∘ 𝐹) = ran (1^st ↾ ran 𝐹)
10		frn 5522	. . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶))
11		ssres2 5070	. . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1^st ↾ ran 𝐹) ⊆ (1^st ↾ (𝐵 × 𝐶)))
12		rnss 4992	. . . . 5 ⊢ ((1^st ↾ ran 𝐹) ⊆ (1^st ↾ (𝐵 × 𝐶)) → ran (1^st ↾ ran 𝐹) ⊆ ran (1^st ↾ (𝐵 × 𝐶)))
13	10, 11, 12	3syl 17	. . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ↾ ran 𝐹) ⊆ ran (1^st ↾ (𝐵 × 𝐶)))
14		f1stres 6366	. . . . 5 ⊢ (1^st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵
15		frn 5522	. . . . 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 3254	. . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ↾ ran 𝐹) ⊆ 𝐵)
18	9, 17	eqsstrid 3288	. 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1^st ∘ 𝐹) ⊆ 𝐵)
19		df-f 5361	. 2 ⊢ ((1^st ∘ 𝐹):𝐴⟶𝐵 ↔ ((1^st ∘ 𝐹) Fn 𝐴 ∧ ran (1^st ∘ 𝐹) ⊆ 𝐵))
20	8, 18, 19	sylanbrc 417	1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^st ∘ 𝐹):𝐴⟶𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 Vcvv 2815 ⊆ wss 3214 × cxp 4752 ran crn 4755 ↾ cres 4756 ∘ ccom 4758 Fn wfn 5352 ⟶wf 5353 –onto→wfo 5355 1^st c1st 6345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fo 5363 df-fv 5365 df-1st 6347

This theorem is referenced by: (None)

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