Theorem refbas 23628

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 23626	. . 3 ⊢ Rel Ref
2	1	brrelex1i 5708	. 2 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
3		refbas.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
4		refbas.2	. . . 4 ⊢ 𝑌 = ∪ 𝐵
5	3, 4	isref 23627	. . 3 ⊢ (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
6	5	simprbda 503	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋)
7	2, 6	mpancom 700	1 ⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  ∪ cuni 4868   class class class wbr 5105  Refcref 23620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-ref 23623

This theorem is referenced by:  reftr  23632  refun0  23633  locfinreflem  34147  cmpcref  34157  cmppcmp  34165  refssfne  36731