Theorem bnj1400 30160

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 5256	. 2 ⊢ dom ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 dom 𝑧
2		df-iun 4457	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥}
3		df-iun 4457	. . . 4 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧}
4		bnj1400.1	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
5	4	nfcii 2742	. . . . . 6 ⊢ Ⅎ𝑥𝐴
6		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑧𝐴
7		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑧 𝑦 ∈ dom 𝑥
8		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ dom 𝑧
9		dmeq 5246	. . . . . . 7 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
10	9	eleq2d 2673	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥 ↔ 𝑦 ∈ dom 𝑧))
11	5, 6, 7, 8, 10	cbvrexf 3142	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧)
12	11	abbii 2726	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧}
13	3, 12	eqtr4i 2635	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥}
14	2, 13	eqtr4i 2635	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = ∪ 𝑧 ∈ 𝐴 dom 𝑧
15	1, 14	eqtr4i 2635	1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Syntax hints:   → wi 4  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  ∪ cuni 4372  ∪ ciun 4455  dom cdm 5038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-dm 5048

This theorem is referenced by:  bnj1398  30356  bnj1450  30372  bnj1498  30383  bnj1501  30389