Theorem uniiunlem 4040

Description: A subset relationship useful for converting union to indexed union using dfiun2 4989 or dfiun2g 4987 and intersection to indexed intersection using dfiin2 4990. (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 2766	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑧 = 𝐵))
2	1	rexbidv 3186	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	cbvabv 2832	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}
4	3	sseq1i 3964	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶)
5		r19.23v 3189	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
6	5	albii 1839	. . . 4 ⊢ (∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
7		ralcom4 3288	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ ∀𝑧∀𝑥 ∈ 𝐴 (𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
8		abss 4015	. . . 4 ⊢ ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶 ↔ ∀𝑧(∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
9	6, 7, 8	3bitr4i 305	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝐶)
10	4, 9	bitr4i 280	. 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶))
11		nfv 1934	. . . . 5 ⊢ Ⅎ𝑧 𝐵 ∈ 𝐶
12		eleq1 2850	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
13	11, 12	ceqsalg 3489	. . . 4 ⊢ (𝐵 ∈ 𝐷 → (∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶))
14	13	ralimi 3099	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → ∀𝑥 ∈ 𝐴 (∀𝑧(𝑧 = 𝐵 → 𝑧 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶))
15		ralbi 3117	. . 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 286	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558   = wceq 1560   ∈ wcel 2142  {cab 2740  ∀wral 3076  ∃wrex 3086   ⊆ wss 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-ss 3921

This theorem is referenced by:  mreiincl  17624  iunopn  22955  sigaclci  34426  rankfilimbi  35394  dihglblem5  41919