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Mirrors > Home > ILE Home > Th. List > ixpssmap2g | GIF version |
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6622 avoids ax-coll 4043. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
ixpssmap2g | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpf 6614 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) | |
2 | 1 | adantl 275 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) |
3 | ixpfn 6598 | . . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴) | |
4 | fndm 5222 | . . . . . . 7 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴) | |
5 | vex 2689 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
6 | 5 | dmex 4805 | . . . . . . 7 ⊢ dom 𝑓 ∈ V |
7 | 4, 6 | eqeltrrdi 2231 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V) |
8 | 3, 7 | syl 14 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V) |
9 | elmapg 6555 | . . . . 5 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)) | |
10 | 8, 9 | sylan2 284 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → (𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)) |
11 | 2, 10 | mpbird 166 | . . 3 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴)) |
12 | 11 | ex 114 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴))) |
13 | 12 | ssrdv 3103 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 ∪ ciun 3813 dom cdm 4539 Fn wfn 5118 ⟶wf 5119 (class class class)co 5774 ↑𝑚 cmap 6542 Xcixp 6592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-map 6544 df-ixp 6593 |
This theorem is referenced by: ixpssmapg 6622 |
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