Theorem reliun 4700

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 3847	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2	1	releqi 4662	. 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵})
3		df-rel 4586	. 2 ⊢ (Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V))
4		abss 3193	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
5		df-rel 4586	. . . . . 6 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
6		dfss2 3113	. . . . . 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 2460	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
9		ralcom4 2731	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
10		r19.23v 2563	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
11	10	albii 1447	. . . 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 1330   ∈ wcel 2125  {cab 2140  ∀wral 2432  ∃wrex 2433  Vcvv 2709   ⊆ wss 3098  ∪ ciun 3845   × cxp 4577  Rel wrel 4584

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-in 3104  df-ss 3111  df-iun 3847  df-rel 4586

This theorem is referenced by:  reluni  4702  eliunxp  4718  opeliunxp2  4719  dfco2  5078  coiun  5088  opeliunxp2f  6175  fisumcom2  11312  fprodcom2fi  11500  reldvg  12995