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Mirrors > Home > MPE Home > Th. List > ixpf | Structured version Visualization version GIF version |
Description: A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.) |
Ref | Expression |
---|---|
ixpf | ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elixp2 8940 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | ssiun2 5052 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
3 | 2 | sseld 3994 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ 𝐵 → (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
4 | 3 | ralimia 3078 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
5 | 4 | anim2i 617 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
6 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
7 | nfiu1 5032 | . . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
8 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
9 | 6, 7, 8 | ffnfvf 7140 | . . . 4 ⊢ (𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
10 | 5, 9 | sylibr 234 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) |
11 | 10 | 3adant1 1129 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) |
12 | 1, 11 | sylbi 217 | 1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∪ ciun 4996 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 Xcixp 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ixp 8937 |
This theorem is referenced by: uniixp 8960 ixpssmap2g 8966 ioorrnopnlem 46260 iunhoiioolem 46631 |
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