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Theorem isref 21533
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 32671. 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 21532 . . . 4 Rel Ref
21brrelex2i 5299 . . 3 (𝐴Ref𝐵𝐵 ∈ V)
32anim2i 603 . 2 ((𝐴𝐶𝐴Ref𝐵) → (𝐴𝐶𝐵 ∈ V))
4 simpl 468 . . 3 ((𝐴𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)) → 𝐴𝐶)
5 simpr 471 . . . . . . 7 ((𝐴𝐶𝑌 = 𝑋) → 𝑌 = 𝑋)
6 isref.2 . . . . . . 7 𝑌 = 𝐵
7 isref.1 . . . . . . 7 𝑋 = 𝐴
85, 6, 73eqtr3g 2828 . . . . . 6 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 = 𝐴)
9 uniexg 7102 . . . . . . 7 (𝐴𝐶 𝐴 ∈ V)
109adantr 466 . . . . . 6 ((𝐴𝐶𝑌 = 𝑋) → 𝐴 ∈ V)
118, 10eqeltrd 2850 . . . . 5 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 ∈ V)
12 uniexb 7120 . . . . 5 (𝐵 ∈ V ↔ 𝐵 ∈ V)
1311, 12sylibr 224 . . . 4 ((𝐴𝐶𝑌 = 𝑋) → 𝐵 ∈ V)
1413adantrr 696 . . 3 ((𝐴𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)) → 𝐵 ∈ V)
154, 14jca 501 . 2 ((𝐴𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)) → (𝐴𝐶𝐵 ∈ V))
16 unieq 4582 . . . . . 6 (𝑎 = 𝐴 𝑎 = 𝐴)
1716, 7syl6eqr 2823 . . . . 5 (𝑎 = 𝐴 𝑎 = 𝑋)
1817eqeq2d 2781 . . . 4 (𝑎 = 𝐴 → ( 𝑏 = 𝑎 𝑏 = 𝑋))
19 raleq 3287 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝑏 𝑥𝑦))
2018, 19anbi12d 616 . . 3 (𝑎 = 𝐴 → (( 𝑏 = 𝑎 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑦) ↔ ( 𝑏 = 𝑋 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑦)))
21 unieq 4582 . . . . . 6 (𝑏 = 𝐵 𝑏 = 𝐵)
2221, 6syl6eqr 2823 . . . . 5 (𝑏 = 𝐵 𝑏 = 𝑌)
2322eqeq1d 2773 . . . 4 (𝑏 = 𝐵 → ( 𝑏 = 𝑋𝑌 = 𝑋))
24 rexeq 3288 . . . . 5 (𝑏 = 𝐵 → (∃𝑦𝑏 𝑥𝑦 ↔ ∃𝑦𝐵 𝑥𝑦))
2524ralbidv 3135 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑦))
2623, 25anbi12d 616 . . 3 (𝑏 = 𝐵 → (( 𝑏 = 𝑋 ∧ ∀𝑥𝐴𝑦𝑏 𝑥𝑦) ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
27 df-ref 21529 . . 3 Ref = {⟨𝑎, 𝑏⟩ ∣ ( 𝑏 = 𝑎 ∧ ∀𝑥𝑎𝑦𝑏 𝑥𝑦)}
2820, 26, 27brabg 5127 . 2 ((𝐴𝐶𝐵 ∈ V) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
293, 15, 28pm5.21nd 803 1 (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  wss 3723   cuni 4574   class class class wbr 4786  Refcref 21526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-ref 21529
This theorem is referenced by:  refbas  21534  refssex  21535  ssref  21536  refref  21537  reftr  21538  refun0  21539  dissnref  21552  reff  30246  locfinreflem  30247  cmpcref  30257  fnessref  32689  refssfne  32690
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