Theorem resixp 7829

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 5362	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐶 → (𝐹 ↾ 𝐵) ∈ V)
2	1	adantl 481	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ V)
3		simpr 476	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶)
4		elixp2 7798	. . . . 5 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶))
5	3, 4	sylib 207	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶))
6	5	simp2d 1067	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐹 Fn 𝐴)
7		simpl 472	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐵 ⊆ 𝐴)
8		fnssres 5918	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
9	6, 7, 8	syl2anc 691	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) Fn 𝐵)
10	5	simp3d 1068	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶)
11		ssralv 3629	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶))
12	7, 10, 11	sylc 63	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶)
13		fvres 6117	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
14	13	eleq1d 2672	. . . 4 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
15	14	ralbiia 2962	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶)
16	12, 15	sylibr 223	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶)
17		elixp2 7798	. 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 1239	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804  Xcixp 7794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ixp 7795

This theorem is referenced by:  resixpfo  7832  ixpfi2  8147  ptrescn  21252  ptuncnv  21420  ptcmplem2  21667