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Mirrors > Home > MPE Home > Th. List > Mathboxes > ixpssmapc | Structured version Visualization version GIF version |
Description: An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
ixpssmapc.x | ⊢ Ⅎ𝑥𝜑 |
ixpssmapc.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
ixpssmapc.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
ixpssmapc | ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑m 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpssmapc.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
2 | ixpssmapc.x | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
3 | ixpssmapc.b | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) | |
4 | 3 | ex 415 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶)) |
5 | 2, 4 | ralrimi 3216 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
6 | iunss 4968 | . . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | |
7 | 5, 6 | sylibr 236 | . . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
8 | 1, 7 | ssexd 5227 | . . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
9 | ixpssmap2g 8490 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ V → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴)) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴)) |
11 | mapss 8452 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) ⊆ (𝐶 ↑m 𝐴)) | |
12 | 1, 7, 11 | syl2anc 586 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) ⊆ (𝐶 ↑m 𝐴)) |
13 | 10, 12 | sstrd 3976 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1780 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 ∪ ciun 4918 (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-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 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-res 5566 df-ima 5567 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-1st 7688 df-2nd 7689 df-map 8407 df-ixp 8461 |
This theorem is referenced by: ioorrnopnlem 42588 |
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