MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isref Structured version   Visualization version   GIF version

Theorem isref 21252
Description: The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 32029. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Hypotheses
Ref Expression
isref.1 𝑋 = 𝐴
isref.2 𝑌 = 𝐵
Assertion
Ref Expression
isref (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵
Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem isref
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 refrel 21251 . . . 4 Rel Ref
21brrelex2i 5129 . . 3 (𝐴Ref𝐵𝐵 ∈ V)
32anim2i 592 . 2 ((𝐴𝐶𝐴Ref𝐵) → (𝐴𝐶𝐵 ∈ V))
4 simpl 473 . . 3 ((𝐴𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)) → 𝐴𝐶)
5 simpr 477 . . . . . . 7 ((𝐴𝐶𝑌 = 𝑋) → 𝑌 = 𝑋)
6 isref.2 . . . . . . 7 𝑌 = 𝐵
7 isref.1 . . . . . . 7 𝑋 = 𝐴
85, 6, 73eqtr3g 2678 . . . . . 6 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 = 𝐴)
9 uniexg 6920 . . . . . . 7 (𝐴𝐶 𝐴 ∈ V)
109adantr 481 . . . . . 6 ((𝐴𝐶𝑌 = 𝑋) → 𝐴 ∈ V)
118, 10eqeltrd 2698 . . . . 5 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 ∈ V)
12 uniexb 6936 . . . . 5 (𝐵 ∈ V ↔ 𝐵 ∈ V)
1311, 12sylibr 224 . . . 4 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 ∈ V)
1413adantrr 752 . . 3 ((𝐴𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)) → 𝐵 ∈ V)
154, 14jca 554 . 2 ((𝐴𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)) → (𝐴𝐶𝐵 ∈ V))
16 unieq 4417 . . . . . 6 (𝑎 = 𝐴 𝑎 = 𝐴)
1716, 7syl6eqr 2673 . . . . 5 (𝑎 = 𝐴 𝑎 = 𝑋)
1817eqeq2d 2631 . . . 4 (𝑎 = 𝐴 → ( 𝑏 = 𝑎 𝑏 = 𝑋))
19 raleq 3131 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥𝑦))
2018, 19anbi12d 746 . . 3 (𝑎 = 𝐴 → (( 𝑏 = 𝑎 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑦) ↔ ( 𝑏 = 𝑋 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑦)))
21 unieq 4417 . . . . . 6 (𝑏 = 𝐵 𝑏 = 𝐵)
2221, 6syl6eqr 2673 . . . . 5 (𝑏 = 𝐵 𝑏 = 𝑌)
2322eqeq1d 2623 . . . 4 (𝑏 = 𝐵 → ( 𝑏 = 𝑋𝑌 = 𝑋))
24 rexeq 3132 . . . . 5 (𝑏 = 𝐵 → (∃𝑦𝑏 𝑥𝑦 ↔ ∃𝑦𝐵 𝑥𝑦))
2524ralbidv 2982 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑦))
2623, 25anbi12d 746 . . 3 (𝑏 = 𝐵 → (( 𝑏 = 𝑋 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑦) ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
27 df-ref 21248 . . 3 Ref = {⟨𝑎, 𝑏⟩ ∣ ( 𝑏 = 𝑎 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑦)}
2820, 26, 27brabg 4964 . 2 ((𝐴𝐶𝐵 ∈ V) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
293, 15, 28pm5.21nd 940 1 (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  Vcvv 3190  wss 3560   cuni 4409   class class class wbr 4623  Refcref 21245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-ref 21248
This theorem is referenced by:  refbas  21253  refssex  21254  ssref  21255  refref  21256  reftr  21257  refun0  21258  dissnref  21271  reff  29730  locfinreflem  29731  cmpcref  29741  fnessref  32047  refssfne  32048
  Copyright terms: Public domain W3C validator