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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 22116 | . . 3 ⊢ Rel Ref | |
2 | 1 | brrelex1i 5608 | . 2 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V) |
3 | refbas.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
4 | refbas.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐵 | |
5 | 3, 4 | isref 22117 | . . 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 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 ∪ cuni 4838 class class class wbr 5066 Refcref 22110 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-ref 22113 |
This theorem is referenced by: reftr 22122 refun0 22123 locfinreflem 31104 cmpcref 31114 cmppcmp 31122 refssfne 33706 |
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