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Mirrors > Home > MPE Home > Th. List > uniiunlem | Structured version Visualization version GIF version |
Description: A subset relationship useful for converting union to indexed union using dfiun2 4554 or dfiun2g 4552 and intersection to indexed intersection using dfiin2 4555. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.) |
Ref | Expression |
---|---|
uniiunlem | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2626 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑧 = 𝐵)) | |
2 | 1 | rexbidv 3052 | . . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)) |
3 | 2 | cbvabv 2747 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} |
4 | 3 | sseq1i 3629 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶) |
5 | r19.23v 3023 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶)) | |
6 | 5 | albii 1747 | . . . 4 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶)) |
7 | ralcom4 3224 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶)) | |
8 | abss 3671 | . . . 4 ⊢ ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶 ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶)) | |
9 | 6, 7, 8 | 3bitr4i 292 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶) |
10 | 4, 9 | bitr4i 267 | . 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶)) |
11 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑧 𝐵 ∈ 𝐶 | |
12 | eleq1 2689 | . . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
13 | 11, 12 | ceqsalg 3230 | . . . 4 ⊢ (𝐵 ∈ 𝐷 → (∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶)) |
14 | 13 | ralimi 2952 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → ∀𝑥 ∈ 𝐴 (∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶)) |
15 | ralbi 3068 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶) → (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)) |
17 | 10, 16 | syl5rbb 273 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-in 3581 df-ss 3588 |
This theorem is referenced by: mreiincl 16256 iunopn 20703 sigaclci 30195 dihglblem5 36587 |
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