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Mirrors > Home > MPE Home > Th. List > resixp | Structured version Visualization version GIF version |
Description: Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.) |
Ref | Expression |
---|---|
resixp | ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resexg 5900 | . . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐶 → (𝐹 ↾ 𝐵) ∈ V) | |
2 | 1 | adantl 484 | . 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ V) |
3 | simpr 487 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) | |
4 | elixp2 8467 | . . . . 5 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶)) | |
5 | 3, 4 | sylib 220 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶)) |
6 | 5 | simp2d 1139 | . . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐹 Fn 𝐴) |
7 | simpl 485 | . . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → 𝐵 ⊆ 𝐴) | |
8 | fnssres 6472 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) | |
9 | 6, 7, 8 | syl2anc 586 | . 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) Fn 𝐵) |
10 | 5 | simp3d 1140 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐶) |
11 | ssralv 4035 | . . . 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 6691 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) | |
14 | 13 | eleq1d 2899 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) |
15 | 14 | ralbiia 3166 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶) |
16 | 12, 15 | sylibr 236 | . 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → ∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐶) |
17 | elixp2 8467 | . 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 1339 | 1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ⊆ wss 3938 ↾ cres 5559 Fn wfn 6352 ‘cfv 6357 Xcixp 8463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-ixp 8464 |
This theorem is referenced by: resixpfo 8502 ixpfi2 8824 ptrescn 22249 ptuncnv 22417 ptcmplem2 22663 |
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