Theorem resixp 7985

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 5477	. . 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 7954	. . . . 5 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶))
5	3, 4	sylib 208	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶))
6	5	simp2d 1094	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐹 Fn 𝐴)
7		simpl 472	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐵 ⊆ 𝐴)
8		fnssres 6042	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
9	6, 7, 8	syl2anc 694	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) Fn 𝐵)
10	5	simp3d 1095	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶)
11		ssralv 3699	. . . 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 6245	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
14	13	eleq1d 2715	. . . 4 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))
15	14	ralbiia 3008	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶)
16	12, 15	sylibr 224	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶)
17		elixp2 7954	. 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 1265	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ⊆ wss 3607   ↾ cres 5145   Fn wfn 5921  ‘cfv 5926  Xcixp 7950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ixp 7951

This theorem is referenced by:  resixpfo  7988  ixpfi2  8305  ptrescn  21490  ptuncnv  21658  ptcmplem2  21904