Theorem refbas 22661

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 22659	. . 3 ⊢ Rel Ref
2	1	brrelex1i 5643	. 2 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
3		refbas.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
4		refbas.2	. . . 4 ⊢ 𝑌 = ∪ 𝐵
5	3, 4	isref 22660	. . 3 ⊢ (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
6	5	simprbda 499	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋)
7	2, 6	mpancom 685	1 ⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074  Refcref 22653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-ref 22656

This theorem is referenced by:  reftr  22665  refun0  22666  locfinreflem  31790  cmpcref  31800  cmppcmp  31808  refssfne  34547