Theorem reluni 3322

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:   x,A

Proof of Theorem reluni

Step	Hyp	Ref	Expression
1		r19.23v 1788	. . . 4 ⊢ (∀x ∈ A (y ∈ x → y ∈ (V × V)) ↔ (∃x ∈ A y ∈ x → y ∈ (V × V)))
2		eluni2 2561	. . . . 5 ⊢ (y ∈ ∪A ↔ ∃x ∈ A y ∈ x)
3	2	imbi1i 193	. . . 4 ⊢ ((y ∈ ∪A → y ∈ (V × V)) ↔ (∃x ∈ A y ∈ x → y ∈ (V × V)))
4	1, 3	bitr4i 183	. . 3 ⊢ (∀x ∈ A (y ∈ x → y ∈ (V × V)) ↔ (y ∈ ∪A → y ∈ (V × V)))
5	4	albii 1040	. 2 ⊢ (∀y∀x ∈ A (y ∈ x → y ∈ (V × V)) ↔ ∀y(y ∈ ∪A → y ∈ (V × V)))
6		df-rel 3242	. . . . 5 ⊢ (Rel x ↔ x ⊆ (V × V))
7		dfss2 2109	. . . . 5 ⊢ (x ⊆ (V × V) ↔ ∀y(y ∈ x → y ∈ (V × V)))
8	6, 7	bitri 180	. . . 4 ⊢ (Rel x ↔ ∀y(y ∈ x → y ∈ (V × V)))
9	8	ralbii 1714	. . 3 ⊢ (∀x ∈ A Rel x ↔ ∀x ∈ A ∀y(y ∈ x → y ∈ (V × V)))
10		ralcom4 1870	. . 3 ⊢ (∀x ∈ A ∀y(y ∈ x → y ∈ (V × V)) ↔ ∀y∀x ∈ A (y ∈ x → y ∈ (V × V)))
11	9, 10	bitri 180	. 2 ⊢ (∀x ∈ A Rel x ↔ ∀y∀x ∈ A (y ∈ x → y ∈ (V × V)))
12		df-rel 3242	. . 3 ⊢ (Rel ∪A ↔ ∪A ⊆ (V × V))
13		dfss2 2109	. . 3 ⊢ (∪A ⊆ (V × V) ↔ ∀y(y ∈ ∪A → y ∈ (V × V)))
14	12, 13	bitri 180	. 2 ⊢ (Rel ∪A ↔ ∀y(y ∈ ∪A → y ∈ (V × V)))
15	5, 11, 14	3bitr4ri 191	1 ⊢ (Rel ∪A ↔ ∀x ∈ A Rel x)

Syntax hints:   → wi 3   ↔ wb 153  ∀wal 995   ∈ wcel 999  ∀wral 1692  ∃wrex 1693  Vcvv 1858   ⊆ wss 2098  ∪cuni 2557   × cxp 3225  Rel wrel 3232

This theorem is referenced by:  fununi 3620  tfrlem6 3974

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-ral 1696  df-rex 1697  df-v 1859  df-in 2102  df-ss 2104  df-uni 2558  df-rel 3242

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