Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 450 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶)) |
5 | 2, 4 | ralrimi 2957 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
6 | iunss 4561 | . . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | |
7 | 5, 6 | sylibr 224 | . . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
8 | 1, 7 | ssexd 4805 | . . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
9 | ixpssmap2g 7937 | . . 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 7900 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴) ⊆ (𝐶 ↑𝑚 𝐴)) | |
12 | 1, 7, 11 | syl2anc 693 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴) ⊆ (𝐶 ↑𝑚 𝐴)) |
13 | 10, 12 | sstrd 3613 | 1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑𝑚 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 ∪ ciun 4520 (class class class)co 6650 ↑𝑚 cmap 7857 Xcixp 7908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-ixp 7909 |
This theorem is referenced by: ioorrnopnlem 40524 |
Copyright terms: Public domain | W3C validator |