Theorem bnj1400 34965

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 5858	. 2 ⊢ dom ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 dom 𝑧
2		df-iun 4925	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥}
3		df-iun 4925	. . . 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 2897	. . . . . 6 ⊢ Ⅎ𝑧𝐴
7		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑧 𝑦 ∈ dom 𝑥
8		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ dom 𝑧
9		dmeq 5847	. . . . . . 7 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
10	9	eleq2d 2821	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥 ↔ 𝑦 ∈ dom 𝑧))
11	5, 6, 7, 8, 10	cbvrexfw 3276	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧)
12	11	abbii 2802	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧}
13	3, 12	eqtr4i 2761	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥}
14	2, 13	eqtr4i 2761	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = ∪ 𝑧 ∈ 𝐴 dom 𝑧
15	1, 14	eqtr4i 2761	1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2713  ∃wrex 3059  ∪ cuni 4840  ∪ ciun 4923  dom cdm 5620

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-dm 5630

This theorem is referenced by:  bnj1398  35164  bnj1450  35180  bnj1498  35191  bnj1501  35197