Theorem resixp 8874

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 5986	. . 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 8842	. . . . 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 6615	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
9	6, 7, 8	syl2anc 585	. 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 3991	. . . 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 6853	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
14	13	eleq1d 2822	. . . 4 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
15	14	ralbiia 3082	. . 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 8842	. 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 1345	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890   ↾ cres 5626   Fn wfn 6487  ‘cfv 6492  Xcixp 8838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ixp 8839

This theorem is referenced by:  resixpfo  8877  ixpfi2  9253  ptrescn  23614  ptuncnv  23782  ptcmplem2  24028