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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 22118 | . . 3 ⊢ Rel Ref | |
2 | 1 | brrelex1i 5610 | . 2 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V) |
3 | refbas.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
4 | refbas.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐵 | |
5 | 3, 4 | isref 22119 | . . 3 ⊢ (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))) |
6 | 5 | simprbda 501 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋) |
7 | 2, 6 | mpancom 686 | 1 ⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 Refcref 22112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-ref 22115 |
This theorem is referenced by: reftr 22124 refun0 22125 locfinreflem 31106 cmpcref 31116 cmppcmp 31124 refssfne 33708 |
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