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Mathbox for Glauco Siliprandi |
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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 414 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶)) |
5 | 2, 4 | ralrimi 3255 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
6 | iunss 5047 | . . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | |
7 | 5, 6 | sylibr 233 | . . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
8 | 1, 7 | ssexd 5323 | . . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
9 | ixpssmap2g 8917 | . . 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 8879 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) ⊆ (𝐶 ↑m 𝐴)) | |
12 | 1, 7, 11 | syl2anc 585 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) ⊆ (𝐶 ↑m 𝐴)) |
13 | 10, 12 | sstrd 3991 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 Ⅎwnf 1786 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ⊆ wss 3947 ∪ ciun 4996 (class class class)co 7404 ↑m cmap 8816 Xcixp 8887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-map 8818 df-ixp 8888 |
This theorem is referenced by: ioorrnopnlem 44955 |
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