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Theorem relexp0eq 40466
 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 ((𝐴𝑈𝐵𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴𝑟0) = (𝐵𝑟0)))

Proof of Theorem relexp0eq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfcleq 2792 . . . 4 ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ ∀𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)))
2 alcom 2160 . . . . 5 (∀𝑦𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑥𝑦(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)))
3 19.3v 1986 . . . . 5 (∀𝑦𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)))
4 ax6ev 1972 . . . . . . . . 9 𝑦 𝑦 = 𝑥
5 pm5.5 365 . . . . . . . . 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))))
64, 5ax-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 1943 . . . . . . . 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 1986 . . . . . . . 8 (∀𝑦(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)))
96, 7, 83bitr4ri 307 . . . . . . 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 pm5.32 577 . . . . . . . . 9 ((𝑦 = 𝑥 → (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))) ↔ ((𝑦 = 𝑥𝑥 ∈ (dom 𝐴 ∪ ran 𝐴)) ↔ (𝑦 = 𝑥𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))))
11 ancom 464 . . . . . . . . . 10 ((𝑦 = 𝑥𝑥 ∈ (dom 𝐴 ∪ ran 𝐴)) ↔ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥))
12 ancom 464 . . . . . . . . . 10 ((𝑦 = 𝑥𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥))
1311, 12bibi12i 343 . . . . . . . . 9 (((𝑦 = 𝑥𝑥 ∈ (dom 𝐴 ∪ ran 𝐴)) ↔ (𝑦 = 𝑥𝑥 ∈ (dom 𝐵 ∪ ran 𝐵))) ↔ ((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
1410, 13bitri 278 . . . . . . . 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)))
1514albii 1821 . . . . . . 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)))
169, 15bitri 278 . . . . . 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)))
1716albii 1821 . . . . 5 (∀𝑥𝑦(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑥𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
182, 3, 173bitr3i 304 . . . 4 (∀𝑥(𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ↔ 𝑥 ∈ (dom 𝐵 ∪ ran 𝐵)) ↔ ∀𝑥𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
191, 18bitri 278 . . 3 ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ ∀𝑥𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
20 eqopab2bw 5401 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)} ↔ ∀𝑥𝑦((𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥) ↔ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)))
21 opabresid 5885 . . . . 5 ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥)}
2221eqcomi 2807 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐴 ∪ ran 𝐴) ∧ 𝑦 = 𝑥)} = ( I ↾ (dom 𝐴 ∪ ran 𝐴))
23 opabresid 5885 . . . . 5 ( I ↾ (dom 𝐵 ∪ ran 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)}
2423eqcomi 2807 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝐵 ∪ ran 𝐵) ∧ 𝑦 = 𝑥)} = ( I ↾ (dom 𝐵 ∪ ran 𝐵))
2522, 24eqeq12i 2813 . . 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)))
2619, 20, 253bitr2i 302 . 2 ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
27 relexp0g 14380 . . 3 (𝐴𝑈 → (𝐴𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
28 relexp0g 14380 . . 3 (𝐵𝑉 → (𝐵𝑟0) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
2927, 28eqeqan12d 2815 . 2 ((𝐴𝑈𝐵𝑉) → ((𝐴𝑟0) = (𝐵𝑟0) ↔ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = ( I ↾ (dom 𝐵 ∪ ran 𝐵))))
3026, 29bitr4id 293 1 ((𝐴𝑈𝐵𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴𝑟0) = (𝐵𝑟0)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ∪ cun 3879  {copab 5093   I cid 5425  dom cdm 5520  ran crn 5521   ↾ cres 5522  (class class class)co 7140  0cc0 10533  ↑𝑟crelexp 14377 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-1cn 10591  ax-icn 10592  ax-addcl 10593  ax-mulcl 10595  ax-i2m1 10601 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-iota 6286  df-fun 6329  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-n0 11893  df-relexp 14378 This theorem is referenced by:  iunrelexp0  40467
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