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Theorem isref 21293
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 32309. 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 21292 . . . 4 Rel Ref
21brrelex2i 5149 . . 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 2677 . . . . . 6 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 = 𝐴)
9 uniexg 6940 . . . . . . 7 (𝐴𝐶 𝐴 ∈ V)
109adantr 481 . . . . . 6 ((𝐴𝐶𝑌 = 𝑋) → 𝐴 ∈ V)
118, 10eqeltrd 2699 . . . . 5 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 ∈ V)
12 uniexb 6958 . . . . 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 4435 . . . . . 6 (𝑎 = 𝐴 𝑎 = 𝐴)
1716, 7syl6eqr 2672 . . . . 5 (𝑎 = 𝐴 𝑎 = 𝑋)
1817eqeq2d 2630 . . . 4 (𝑎 = 𝐴 → ( 𝑏 = 𝑎 𝑏 = 𝑋))
19 raleq 3133 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥𝑦))
2018, 19anbi12d 746 . . 3 (𝑎 = 𝐴 → (( 𝑏 = 𝑎 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑦) ↔ ( 𝑏 = 𝑋 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑦)))
21 unieq 4435 . . . . . 6 (𝑏 = 𝐵 𝑏 = 𝐵)
2221, 6syl6eqr 2672 . . . . 5 (𝑏 = 𝐵 𝑏 = 𝑌)
2322eqeq1d 2622 . . . 4 (𝑏 = 𝐵 → ( 𝑏 = 𝑋𝑌 = 𝑋))
24 rexeq 3134 . . . . 5 (𝑏 = 𝐵 → (∃𝑦𝑏 𝑥𝑦 ↔ ∃𝑦𝐵 𝑥𝑦))
2524ralbidv 2983 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑦))
2623, 25anbi12d 746 . . 3 (𝑏 = 𝐵 → (( 𝑏 = 𝑋 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑦) ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
27 df-ref 21289 . . 3 Ref = {⟨𝑎, 𝑏⟩ ∣ ( 𝑏 = 𝑎 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑦)}
2820, 26, 27brabg 4984 . 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 1481  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  wss 3567   cuni 4427   class class class wbr 4644  Refcref 21286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-ref 21289
This theorem is referenced by:  refbas  21294  refssex  21295  ssref  21296  refref  21297  reftr  21298  refun0  21299  dissnref  21312  reff  29880  locfinreflem  29881  cmpcref  29891  fnessref  32327  refssfne  32328
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