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Theorem reftr 22129
 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 2798 . . . 4 𝐵 = 𝐵
2 eqid 2798 . . . 4 𝐶 = 𝐶
31, 2refbas 22125 . . 3 (𝐵Ref𝐶 𝐶 = 𝐵)
4 eqid 2798 . . . 4 𝐴 = 𝐴
54, 1refbas 22125 . . 3 (𝐴Ref𝐵 𝐵 = 𝐴)
63, 5sylan9eqr 2855 . 2 ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐶 = 𝐴)
7 refssex 22126 . . . . . 6 ((𝐴Ref𝐵𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
87ex 416 . . . . 5 (𝐴Ref𝐵 → (𝑥𝐴 → ∃𝑦𝐵 𝑥𝑦))
98adantr 484 . . . 4 ((𝐴Ref𝐵𝐵Ref𝐶) → (𝑥𝐴 → ∃𝑦𝐵 𝑥𝑦))
10 refssex 22126 . . . . . . 7 ((𝐵Ref𝐶𝑦𝐵) → ∃𝑧𝐶 𝑦𝑧)
1110ad2ant2lr 747 . . . . . 6 (((𝐴Ref𝐵𝐵Ref𝐶) ∧ (𝑦𝐵𝑥𝑦)) → ∃𝑧𝐶 𝑦𝑧)
12 sstr2 3922 . . . . . . . 8 (𝑥𝑦 → (𝑦𝑧𝑥𝑧))
1312reximdv 3232 . . . . . . 7 (𝑥𝑦 → (∃𝑧𝐶 𝑦𝑧 → ∃𝑧𝐶 𝑥𝑧))
1413ad2antll 728 . . . . . 6 (((𝐴Ref𝐵𝐵Ref𝐶) ∧ (𝑦𝐵𝑥𝑦)) → (∃𝑧𝐶 𝑦𝑧 → ∃𝑧𝐶 𝑥𝑧))
1511, 14mpd 15 . . . . 5 (((𝐴Ref𝐵𝐵Ref𝐶) ∧ (𝑦𝐵𝑥𝑦)) → ∃𝑧𝐶 𝑥𝑧)
1615rexlimdvaa 3244 . . . 4 ((𝐴Ref𝐵𝐵Ref𝐶) → (∃𝑦𝐵 𝑥𝑦 → ∃𝑧𝐶 𝑥𝑧))
179, 16syld 47 . . 3 ((𝐴Ref𝐵𝐵Ref𝐶) → (𝑥𝐴 → ∃𝑧𝐶 𝑥𝑧))
1817ralrimiv 3148 . 2 ((𝐴Ref𝐵𝐵Ref𝐶) → ∀𝑥𝐴𝑧𝐶 𝑥𝑧)
19 refrel 22123 . . . . 5 Rel Ref
2019brrelex1i 5573 . . . 4 (𝐴Ref𝐵𝐴 ∈ V)
2120adantr 484 . . 3 ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴 ∈ V)
224, 2isref 22124 . . 3 (𝐴 ∈ V → (𝐴Ref𝐶 ↔ ( 𝐶 = 𝐴 ∧ ∀𝑥𝐴𝑧𝐶 𝑥𝑧)))
2321, 22syl 17 . 2 ((𝐴Ref𝐵𝐵Ref𝐶) → (𝐴Ref𝐶 ↔ ( 𝐶 = 𝐴 ∧ ∀𝑥𝐴𝑧𝐶 𝑥𝑧)))
246, 18, 23mpbir2and 712 1 ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴Ref𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∪ cuni 4801   class class class wbr 5031  Refcref 22117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-ref 22120 This theorem is referenced by:  refssfne  33834
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