Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ixpssmap2g | Structured version Visualization version GIF version |
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8491 avoids ax-rep 5189. (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 8483 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) | |
2 | 1 | adantl 484 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) |
3 | n0i 4298 | . . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ¬ X𝑥 ∈ 𝐴 𝐵 = ∅) | |
4 | ixpprc 8482 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅) | |
5 | 3, 4 | nsyl2 143 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐴 ∈ V) |
6 | elmapg 8418 | . . . . 5 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)) | |
7 | 5, 6 | sylan2 594 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → (𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵)) |
8 | 2, 7 | mpbird 259 | . . 3 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴)) |
9 | 8 | ex 415 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴))) |
10 | 9 | ssrdv 3972 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 ∪ ciun 4918 ⟶wf 6350 (class class class)co 7155 ↑m cmap 8405 Xcixp 8460 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-ixp 8461 |
This theorem is referenced by: ixpssmapg 8491 ixpfi 8820 ixpiunwdom 9054 prdsval 16727 prdsbas 16729 ixpssmapc 41334 |
Copyright terms: Public domain | W3C validator |