Theorem resixp 8981

Description: Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)

Assertion

Ref	Expression
resixp	⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem resixp

Step	Hyp	Ref	Expression
1		resexg 6052	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐶 → (𝐹 ↾ 𝐵) ∈ V)
2	1	adantl 481	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ V)
3		simpr 484	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶)
4		elixp2 8949	. . . . 5 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶))
5	3, 4	sylib 218	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶))
6	5	simp2d 1144	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐹 Fn 𝐴)
7		simpl 482	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐵 ⊆ 𝐴)
8		fnssres 6699	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
9	6, 7, 8	syl2anc 584	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) Fn 𝐵)
10	5	simp3d 1145	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶)
11		ssralv 4067	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶))
12	7, 10, 11	sylc 65	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶)
13		fvres 6933	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
14	13	eleq1d 2826	. . . 4 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
15	14	ralbiia 3091	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶)
16	12, 15	sylibr 234	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶)
17		elixp2 8949	. 2 ⊢ ((𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶 ↔ ((𝐹 ↾ 𝐵) ∈ V ∧ (𝐹 ↾ 𝐵) Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶))
18	2, 9, 16, 17	syl3anbrc 1344	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3061  Vcvv 3481   ⊆ wss 3966   ↾ cres 5695   Fn wfn 6564  ‘cfv 6569  Xcixp 8945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-res 5705  df-iota 6522  df-fun 6571  df-fn 6572  df-fv 6577  df-ixp 8946

This theorem is referenced by:  resixpfo  8984  ixpfi2  9397  ptrescn  23672  ptuncnv  23840  ptcmplem2  24086