Theorem refbas 23485

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 23483	. . 3 ⊢ Rel Ref
2	1	brrelex1i 5680	. 2 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
3		refbas.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
4		refbas.2	. . . 4 ⊢ 𝑌 = ∪ 𝐵
5	3, 4	isref 23484	. . 3 ⊢ (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
6	5	simprbda 498	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋)
7	2, 6	mpancom 689	1 ⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  ∪ cuni 4851   class class class wbr 5086  Refcref 23477

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-ref 23480

This theorem is referenced by:  reftr  23489  refun0  23490  locfinreflem  34000  cmpcref  34010  cmppcmp  34018  refssfne  36556