Theorem reluni 2493

Description: Union law for relations. Exercise 6 of [TakeutiZaring] p. 25 and its converse.

Assertion

Ref	Expression
reluni	⊢ (Rel ∪A ↔ ∀x ∈ A Rel x)

Distinct variable group(s):   x,A

Proof of Theorem reluni

Step	Hyp	Ref	Expression
1		r19.23v 1282	. . . 4 ⊢ (∀x ∈ A (y ∈ x → y ∈ (V × V)) ↔ (∃x ∈ A y ∈ x → y ∈ (V × V)))
2		eluni2 1923	. . . . 5 ⊢ (y ∈ ∪A ↔ ∃x ∈ A y ∈ x)
3	2	imbi1i 161	. . . 4 ⊢ ((y ∈ ∪A → y ∈ (V × V)) ↔ (∃x ∈ A y ∈ x → y ∈ (V × V)))
4	1, 3	bitr4 154	. . 3 ⊢ (∀x ∈ A (y ∈ x → y ∈ (V × V)) ↔ (y ∈ ∪A → y ∈ (V × V)))
5	4	bial 695	. 2 ⊢ (∀y∀x ∈ A (y ∈ x → y ∈ (V × V)) ↔ ∀y(y ∈ ∪A → y ∈ (V × V)))
6		df-rel 2425	. . . . 5 ⊢ (Rel x ↔ x ⊆ (V × V))
7		dfss2 1497	. . . . 5 ⊢ (x ⊆ (V × V) ↔ ∀y(y ∈ x → y ∈ (V × V)))
8	6, 7	bitr 151	. . . 4 ⊢ (Rel x ↔ ∀y(y ∈ x → y ∈ (V × V)))
9	8	biral 1223	. . 3 ⊢ (∀x ∈ A Rel x ↔ ∀x ∈ A ∀y(y ∈ x → y ∈ (V × V)))
10		ralcom4 1360	. . 3 ⊢ (∀x ∈ A ∀y(y ∈ x → y ∈ (V × V)) ↔ ∀y∀x ∈ A (y ∈ x → y ∈ (V × V)))
11	9, 10	bitr 151	. 2 ⊢ (∀x ∈ A Rel x ↔ ∀y∀x ∈ A (y ∈ x → y ∈ (V × V)))
12		df-rel 2425	. . 3 ⊢ (Rel ∪A ↔ ∪A ⊆ (V × V))
13		dfss2 1497	. . 3 ⊢ (∪A ⊆ (V × V) ↔ ∀y(y ∈ ∪A → y ∈ (V × V)))
14	12, 13	bitr 151	. 2 ⊢ (Rel ∪A ↔ ∀y(y ∈ ∪A → y ∈ (V × V)))
15	5, 11, 14	3bitr4r 159	1 ⊢ (Rel ∪A ↔ ∀x ∈ A Rel x)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wel 803   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ∪cuni 1919   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  fununi 2705  tfrlem6 2954

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920  df-rel 2425

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