Theorem uniiunlem 4084

Description: A subset relationship useful for converting union to indexed union using dfiun2 5036 or dfiun2g 5033 and intersection to indexed intersection using dfiin2 5037. (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 2736	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑧 = 𝐵))
2	1	rexbidv 3178	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	cbvabv 2805	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}
4	3	sseq1i 4010	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶)
5		r19.23v 3182	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
6	5	albii 1821	. . . 4 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
7		ralcom4 3283	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
8		abss 4057	. . . 4 ⊢ ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶 ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
9	6, 7, 8	3bitr4i 302	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶)
10	4, 9	bitr4i 277	. 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
11		nfv 1917	. . . . 5 ⊢ Ⅎ𝑧 𝐵 ∈ 𝐶
12		eleq1 2821	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
13	11, 12	ceqsalg 3507	. . . 4 ⊢ (𝐵 ∈ 𝐷 → (∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶))
14	13	ralimi 3083	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → ∀𝑥 ∈ 𝐴 (∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶))
15		ralbi 3103	. . 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	bitr2id 283	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070   ⊆ wss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-v 3476  df-in 3955  df-ss 3965

This theorem is referenced by:  mreiincl  17539  iunopn  22399  sigaclci  33125  dihglblem5  40164