Theorem reliun 4660

Description: An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)

Assertion

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

Proof of Theorem reliun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3815	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2	1	releqi 4622	. 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵})
3		df-rel 4546	. 2 ⊢ (Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V))
4		abss 3166	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
5		df-rel 4546	. . . . . 6 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
6		dfss2 3086	. . . . . 6 ⊢ (𝐵 ⊆ (V × V) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
7	5, 6	bitri 183	. . . . 5 ⊢ (Rel 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
8	7	ralbii 2441	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
9		ralcom4 2708	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
10		r19.23v 2541	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
11	10	albii 1446	. . . 4 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
12	8, 9, 11	3bitri 205	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
13	4, 12	bitr4i 186	. 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V) ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵)
14	2, 3, 13	3bitri 205	1 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417  Vcvv 2686   ⊆ wss 3071  ∪ ciun 3813   × cxp 4537  Rel wrel 4544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-iun 3815  df-rel 4546

This theorem is referenced by:  reluni  4662  eliunxp  4678  opeliunxp2  4679  dfco2  5038  coiun  5048  opeliunxp2f  6135  fisumcom2  11207  reldvg  12817