Theorem bnj1400 34311

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1400.1	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Assertion

Ref	Expression
bnj1400	⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bnj1400

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmuni 5914	. 2 ⊢ dom ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 dom 𝑧
2		df-iun 4999	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥}
3		df-iun 4999	. . . 4 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧}
4		bnj1400.1	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
5	4	nfcii 2886	. . . . . 6 ⊢ Ⅎ𝑥𝐴
6		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑧𝐴
7		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑧 𝑦 ∈ dom 𝑥
8		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ dom 𝑧
9		dmeq 5903	. . . . . . 7 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
10	9	eleq2d 2818	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥 ↔ 𝑦 ∈ dom 𝑧))
11	5, 6, 7, 8, 10	cbvrexfw 3301	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧)
12	11	abbii 2801	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧}
13	3, 12	eqtr4i 2762	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥}
14	2, 13	eqtr4i 2762	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = ∪ 𝑧 ∈ 𝐴 dom 𝑧
15	1, 14	eqtr4i 2762	1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540   ∈ wcel 2105  {cab 2708  ∃wrex 3069  ∪ cuni 4908  ∪ ciun 4997  dom cdm 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-dm 5686

This theorem is referenced by:  bnj1398  34510  bnj1450  34526  bnj1498  34537  bnj1501  34543