Theorem reliunOLD 5763

Description: Obsolete version of reliun 5762 as of 2-Feb-2025. (Contributed by NM, 19-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem reliunOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4945	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2	1	releqi 5724	. 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵})
3		df-rel 5628	. 2 ⊢ (Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V))
4		abss 4012	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
5		df-rel 5628	. . . . . 6 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
6		df-ss 3916	. . . . . 6 ⊢ (𝐵 ⊆ (V × V) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
7	5, 6	bitri 275	. . . . 5 ⊢ (Rel 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
8	7	ralbii 3080	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
9		ralcom4 3260	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
10		r19.23v 3161	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
11	10	albii 1820	. . . 4 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
12	8, 9, 11	3bitri 297	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)))
13	4, 12	bitr4i 278	. 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V) ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵)
14	2, 3, 13	3bitri 297	1 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   ∈ wcel 2113  {cab 2711  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ⊆ wss 3899  ∪ ciun 4943   × cxp 5619  Rel wrel 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ral 3050  df-rex 3059  df-ss 3916  df-iun 4945  df-rel 5628

This theorem is referenced by: (None)