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| Description: A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) | 
| Ref | Expression | 
|---|---|
| refbas.1 | ⊢ 𝑋 = ∪ 𝐴 | 
| refbas.2 | ⊢ 𝑌 = ∪ 𝐵 | 
| Ref | Expression | 
|---|---|
| refbas | ⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | refrel 23516 | . . 3 ⊢ Rel Ref | |
| 2 | 1 | brrelex1i 5741 | . 2 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V) | 
| 3 | refbas.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
| 4 | refbas.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐵 | |
| 5 | 3, 4 | isref 23517 | . . 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 688 | 1 ⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 Refcref 23510 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-ref 23513 | 
| This theorem is referenced by: reftr 23522 refun0 23523 locfinreflem 33839 cmpcref 33849 cmppcmp 33857 refssfne 36359 | 
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