Theorem refbas 22215

Description: A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Hypotheses

Ref	Expression
refbas.1	⊢ 𝑋 = ∪ 𝐴
refbas.2	⊢ 𝑌 = ∪ 𝐵

Assertion

Ref	Expression
refbas	⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋)

Proof of Theorem refbas

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

Step	Hyp	Ref	Expression
1		refrel 22213	. . 3 ⊢ Rel Ref
2	1	brrelex1i 5581	. 2 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
3		refbas.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
4		refbas.2	. . . 4 ⊢ 𝑌 = ∪ 𝐵
5	3, 4	isref 22214	. . 3 ⊢ (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
6	5	simprbda 502	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋)
7	2, 6	mpancom 687	1 ⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ⊆ wss 3860  ∪ cuni 4801   class class class wbr 5035  Refcref 22207

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 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-ref 22210

This theorem is referenced by:  reftr  22219  refun0  22220  locfinreflem  31315  cmpcref  31325  cmppcmp  31333  refssfne  34122