Theorem ss2iundv 44312

Description: Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)

Hypotheses

Ref	Expression
ss2iundv.el	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)
ss2iundv.sub	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
ss2iundv.ss	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺)

Assertion

Ref	Expression
ss2iundv	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)

Distinct variable groups:   𝑥,𝑦,𝜑   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷   𝑦,𝐺   𝑦,𝑌

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐷(𝑦)   𝐺(𝑥)   𝑌(𝑥)

Proof of Theorem ss2iundv

Step	Hyp	Ref	Expression
1		nfv 1941	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1941	. 2 ⊢ Ⅎ𝑦𝜑
3		nfcv 2931	. 2 ⊢ Ⅎ𝑦𝑌
4		nfcv 2931	. 2 ⊢ Ⅎ𝑦𝐴
5		nfcv 2931	. 2 ⊢ Ⅎ𝑦𝐵
6		nfcv 2931	. 2 ⊢ Ⅎ𝑥𝐶
7		nfcv 2931	. 2 ⊢ Ⅎ𝑦𝐶
8		nfcv 2931	. 2 ⊢ Ⅎ𝑥𝐷
9		nfcv 2931	. 2 ⊢ Ⅎ𝑦𝐺
10		ss2iundv.el	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)
11		ss2iundv.sub	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
12		ss2iundv.ss	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ss2iundf 44311	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  ∪ ciun 4960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-v 3465  df-ss 3930  df-iun 4962

This theorem is referenced by: (None)