Theorem bnj1400 34818

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 5868	. 2 ⊢ dom ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 dom 𝑧
2		df-iun 4953	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥}
3		df-iun 4953	. . . 4 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧}
4		bnj1400.1	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
5	4	nfcii 2880	. . . . . 6 ⊢ Ⅎ𝑥𝐴
6		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑧𝐴
7		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑧 𝑦 ∈ dom 𝑥
8		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ dom 𝑧
9		dmeq 5857	. . . . . . 7 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
10	9	eleq2d 2814	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥 ↔ 𝑦 ∈ dom 𝑧))
11	5, 6, 7, 8, 10	cbvrexfw 3277	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧)
12	11	abbii 2796	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧}
13	3, 12	eqtr4i 2755	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥}
14	2, 13	eqtr4i 2755	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = ∪ 𝑧 ∈ 𝐴 dom 𝑧
15	1, 14	eqtr4i 2755	1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  ∪ cuni 4867  ∪ ciun 4951  dom cdm 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-dm 5641

This theorem is referenced by:  bnj1398  35017  bnj1450  35033  bnj1498  35044  bnj1501  35050