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Theorem isref 23533
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 36322. 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 23532 . . . 4 Rel Ref
21brrelex2i 5746 . . 3 (𝐴Ref𝐵𝐵 ∈ V)
32anim2i 617 . 2 ((𝐴𝐶𝐴Ref𝐵) → (𝐴𝐶𝐵 ∈ V))
4 simpl 482 . . 3 ((𝐴𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)) → 𝐴𝐶)
5 simpr 484 . . . . . . 7 ((𝐴𝐶𝑌 = 𝑋) → 𝑌 = 𝑋)
6 isref.2 . . . . . . 7 𝑌 = 𝐵
7 isref.1 . . . . . . 7 𝑋 = 𝐴
85, 6, 73eqtr3g 2798 . . . . . 6 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 = 𝐴)
9 uniexg 7759 . . . . . . 7 (𝐴𝐶 𝐴 ∈ V)
109adantr 480 . . . . . 6 ((𝐴𝐶𝑌 = 𝑋) → 𝐴 ∈ V)
118, 10eqeltrd 2839 . . . . 5 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 ∈ V)
12 uniexb 7783 . . . . 5 (𝐵 ∈ V ↔ 𝐵 ∈ V)
1311, 12sylibr 234 . . . 4 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 ∈ V)
1413adantrr 717 . . 3 ((𝐴𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)) → 𝐵 ∈ V)
154, 14jca 511 . 2 ((𝐴𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)) → (𝐴𝐶𝐵 ∈ V))
16 unieq 4923 . . . . . 6 (𝑎 = 𝐴 𝑎 = 𝐴)
1716, 7eqtr4di 2793 . . . . 5 (𝑎 = 𝐴 𝑎 = 𝑋)
1817eqeq2d 2746 . . . 4 (𝑎 = 𝐴 → ( 𝑏 = 𝑎 𝑏 = 𝑋))
19 raleq 3321 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥𝑦))
2018, 19anbi12d 632 . . 3 (𝑎 = 𝐴 → (( 𝑏 = 𝑎 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑦) ↔ ( 𝑏 = 𝑋 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑦)))
21 unieq 4923 . . . . . 6 (𝑏 = 𝐵 𝑏 = 𝐵)
2221, 6eqtr4di 2793 . . . . 5 (𝑏 = 𝐵 𝑏 = 𝑌)
2322eqeq1d 2737 . . . 4 (𝑏 = 𝐵 → ( 𝑏 = 𝑋𝑌 = 𝑋))
24 rexeq 3320 . . . . 5 (𝑏 = 𝐵 → (∃𝑦𝑏 𝑥𝑦 ↔ ∃𝑦𝐵 𝑥𝑦))
2524ralbidv 3176 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑦))
2623, 25anbi12d 632 . . 3 (𝑏 = 𝐵 → (( 𝑏 = 𝑋 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑦) ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
27 df-ref 23529 . . 3 Ref = {⟨𝑎, 𝑏⟩ ∣ ( 𝑏 = 𝑎 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑦)}
2820, 26, 27brabg 5549 . 2 ((𝐴𝐶𝐵 ∈ V) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
293, 15, 28pm5.21nd 802 1 (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963   cuni 4912   class class class wbr 5148  Refcref 23526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-ref 23529
This theorem is referenced by:  refbas  23534  refssex  23535  ssref  23536  refref  23537  reftr  23538  refun0  23539  dissnref  23552  reff  33800  locfinreflem  33801  cmpcref  33811  fnessref  36340  refssfne  36341
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