Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunssf | Structured version Visualization version GIF version |
Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
iunssf.1 | ⊢ Ⅎ𝑥𝐶 |
Ref | Expression |
---|---|
iunssf | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 4921 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
2 | 1 | sseq1i 3995 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ 𝐶) |
3 | abss 4040 | . 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ 𝐶 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) | |
4 | dfss2 3955 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) | |
5 | 4 | ralbii 3165 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
6 | ralcom4 3235 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) | |
7 | iunssf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
8 | 7 | nfcri 2971 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶 |
9 | 8 | r19.23 3314 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
10 | 9 | albii 1820 | . . 3 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
11 | 5, 6, 10 | 3bitrri 300 | . 2 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
12 | 2, 3, 11 | 3bitri 299 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 {cab 2799 Ⅎwnfc 2961 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 ∪ ciun 4919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-in 3943 df-ss 3952 df-iun 4921 |
This theorem is referenced by: iunmapss 41498 |
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