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Mirrors > Home > MPE Home > Th. List > refbas | Structured version Visualization version GIF version |
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 21121 | . . 3 ⊢ Rel Ref | |
2 | 1 | brrelexi 5082 | . 2 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V) |
3 | refbas.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
4 | refbas.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐵 | |
5 | 3, 4 | isref 21122 | . . 3 ⊢ (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))) |
6 | 5 | simprbda 651 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋) |
7 | 2, 6 | mpancom 700 | 1 ⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 Refcref 21115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-ref 21118 |
This theorem is referenced by: reftr 21127 refun0 21128 locfinreflem 29235 cmpcref 29245 cmppcmp 29253 refssfne 31523 |
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