Theorem uniiunlem 3268

Description: A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Assertion

Ref	Expression
uniiunlem	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem uniiunlem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2200	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑧 = 𝐵))
2	1	rexbidv 2495	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	cbvabv 2318	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}
4	3	sseq1i 3205	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶)
5		r19.23v 2603	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
6	5	albii 1481	. . . 4 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
7		ralcom4 2782	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
8		abss 3248	. . . 4 ⊢ ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶 ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
9	6, 7, 8	3bitr4i 212	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶)
10	4, 9	bitr4i 187	. 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
11		nfv 1539	. . . . 5 ⊢ Ⅎ𝑧 𝐵 ∈ 𝐶
12		eleq1 2256	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
13	11, 12	ceqsalg 2788	. . . 4 ⊢ (𝐵 ∈ 𝐷 → (∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶))
14	13	ralimi 2557	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → ∀𝑥 ∈ 𝐴 (∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶))
15		ralbi 2626	. . 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 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶))
17	10, 16	bitr2id 193	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473   ⊆ wss 3153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166

This theorem is referenced by:  iunopn  14170