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Theorem reftr 23522
Description: Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Assertion
Ref Expression
reftr ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴Ref𝐶)

Proof of Theorem reftr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 𝐵 = 𝐵
2 eqid 2737 . . . 4 𝐶 = 𝐶
31, 2refbas 23518 . . 3 (𝐵Ref𝐶 𝐶 = 𝐵)
4 eqid 2737 . . . 4 𝐴 = 𝐴
54, 1refbas 23518 . . 3 (𝐴Ref𝐵 𝐵 = 𝐴)
63, 5sylan9eqr 2799 . 2 ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐶 = 𝐴)
7 refssex 23519 . . . . . 6 ((𝐴Ref𝐵𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
87ex 412 . . . . 5 (𝐴Ref𝐵 → (𝑥𝐴 → ∃𝑦𝐵 𝑥𝑦))
98adantr 480 . . . 4 ((𝐴Ref𝐵𝐵Ref𝐶) → (𝑥𝐴 → ∃𝑦𝐵 𝑥𝑦))
10 refssex 23519 . . . . . . 7 ((𝐵Ref𝐶𝑦𝐵) → ∃𝑧𝐶 𝑦𝑧)
1110ad2ant2lr 748 . . . . . 6 (((𝐴Ref𝐵𝐵Ref𝐶) ∧ (𝑦𝐵𝑥𝑦)) → ∃𝑧𝐶 𝑦𝑧)
12 sstr2 3990 . . . . . . . 8 (𝑥𝑦 → (𝑦𝑧𝑥𝑧))
1312reximdv 3170 . . . . . . 7 (𝑥𝑦 → (∃𝑧𝐶 𝑦𝑧 → ∃𝑧𝐶 𝑥𝑧))
1413ad2antll 729 . . . . . 6 (((𝐴Ref𝐵𝐵Ref𝐶) ∧ (𝑦𝐵𝑥𝑦)) → (∃𝑧𝐶 𝑦𝑧 → ∃𝑧𝐶 𝑥𝑧))
1511, 14mpd 15 . . . . 5 (((𝐴Ref𝐵𝐵Ref𝐶) ∧ (𝑦𝐵𝑥𝑦)) → ∃𝑧𝐶 𝑥𝑧)
1615rexlimdvaa 3156 . . . 4 ((𝐴Ref𝐵𝐵Ref𝐶) → (∃𝑦𝐵 𝑥𝑦 → ∃𝑧𝐶 𝑥𝑧))
179, 16syld 47 . . 3 ((𝐴Ref𝐵𝐵Ref𝐶) → (𝑥𝐴 → ∃𝑧𝐶 𝑥𝑧))
1817ralrimiv 3145 . 2 ((𝐴Ref𝐵𝐵Ref𝐶) → ∀𝑥𝐴𝑧𝐶 𝑥𝑧)
19 refrel 23516 . . . . 5 Rel Ref
2019brrelex1i 5741 . . . 4 (𝐴Ref𝐵𝐴 ∈ V)
2120adantr 480 . . 3 ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴 ∈ V)
224, 2isref 23517 . . 3 (𝐴 ∈ V → (𝐴Ref𝐶 ↔ ( 𝐶 = 𝐴 ∧ ∀𝑥𝐴𝑧𝐶 𝑥𝑧)))
2321, 22syl 17 . 2 ((𝐴Ref𝐵𝐵Ref𝐶) → (𝐴Ref𝐶 ↔ ( 𝐶 = 𝐴 ∧ ∀𝑥𝐴𝑧𝐶 𝑥𝑧)))
246, 18, 23mpbir2and 713 1 ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴Ref𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951   cuni 4907   class class class wbr 5143  Refcref 23510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-ref 23513
This theorem is referenced by:  refssfne  36359
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