Theorem relexp0eq 43683

Description: The zeroth power of relationships is the same if and only if the union of their domain and ranges is the same. (Contributed by RP, 11-Jun-2020.)

Assertion

Ref	Expression
relexp0eq	⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴↑𝑟0) = (𝐵↑𝑟0)))

Proof of Theorem relexp0eq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2722	. . . 4 ⊢ ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ ∀𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)))
2		alcom 2160	. . . . 5 ⊢ (∀𝑦∀𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑥∀𝑦(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)))
3		19.3v 1982	. . . . 5 ⊢ (∀𝑦∀𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)))
4		ax6ev 1969	. . . . . . . . 9 ⊢ ∃𝑦 𝑦 = 𝑥
5		pm5.5 361	. . . . . . . . 9 ⊢ (∃𝑦 𝑦 = 𝑥 → ((∃𝑦 𝑦 = 𝑥 → (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))) ↔ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))))
6	4, 5	ax-mp 5	. . . . . . . 8 ⊢ ((∃𝑦 𝑦 = 𝑥 → (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))) ↔ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)))
7		19.23v 1942	. . . . . . . 8 ⊢ (∀𝑦(𝑦 = 𝑥 → (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))) ↔ (∃𝑦 𝑦 = 𝑥 → (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))))
8		19.3v 1982	. . . . . . . 8 ⊢ (∀𝑦(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)))
9	6, 7, 8	3bitr4ri 304	. . . . . . 7 ⊢ (∀𝑦(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))))
10		pm5.32 573	. . . . . . . . 9 ⊢ ((𝑦 = 𝑥 → (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))) ↔ ((𝑦 = 𝑥 ∧ 𝑥 ∈ (dom 𝐴 ∪ ran 𝐴)) ↔ (𝑦 = 𝑥 ∧ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))))
11		ancom 460	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 ∈ (dom 𝐴 ∪ ran 𝐴)) ↔ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥))
12		ancom 460	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥))
13	11, 12	bibi12i 339	. . . . . . . . 9 ⊢ (((𝑦 = 𝑥 ∧ 𝑥 ∈ (dom 𝐴 ∪ ran 𝐴)) ↔ (𝑦 = 𝑥 ∧ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))) ↔ ((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
14	10, 13	bitri 275	. . . . . . . 8 ⊢ ((𝑦 = 𝑥 → (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))) ↔ ((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
15	14	albii 1819	. . . . . . 7 ⊢ (∀𝑦(𝑦 = 𝑥 → (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))) ↔ ∀𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
16	9, 15	bitri 275	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
17	16	albii 1819	. . . . 5 ⊢ (∀𝑥∀𝑦(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑥∀𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
18	2, 3, 17	3bitr3i 301	. . . 4 ⊢ (∀𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑥∀𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
19	1, 18	bitri 275	. . 3 ⊢ ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ ∀𝑥∀𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
20		eqopab2bw 5503	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)} ↔ ∀𝑥∀𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
21		opabresid 6010	. . . . 5 ⊢ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥)}
22	21	eqcomi 2738	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥)} = ( I ↾ (dom 𝐴 ∪ ran 𝐴))
23		opabresid 6010	. . . . 5 ⊢ ( I ↾ (dom 𝐵 ∪ ran 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)}
24	23	eqcomi 2738	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)} = ( I ↾ (dom 𝐵 ∪ ran 𝐵))
25	22, 24	eqeq12i 2747	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)} ↔ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
26	19, 20, 25	3bitr2i 299	. 2 ⊢ ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
27		relexp0g 14964	. . 3 ⊢ (𝐴 ∈ 𝑈 → (𝐴↑𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
28		relexp0g 14964	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵↑𝑟0) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
29	27, 28	eqeqan12d 2743	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((𝐴↑𝑟0) = (𝐵↑𝑟0) ↔ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = ( I ↾ (dom 𝐵 ∪ ran 𝐵))))
30	26, 29	bitr4id 290	1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴↑𝑟0) = (𝐵↑𝑟0)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ∪ cun 3909  {copab 5164   I cid 5525  dom cdm 5631  ran crn 5632   ↾ cres 5633  (class class class)co 7369  0cc0 11044  ↑_𝑟crelexp 14961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-mulcl 11106  ax-i2m1 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-n0 12419  df-relexp 14962

This theorem is referenced by:  iunrelexp0  43684