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Theorem reftr 21127
 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 2610 . . . 4 𝐵 = 𝐵
2 eqid 2610 . . . 4 𝐶 = 𝐶
31, 2refbas 21123 . . 3 (𝐵Ref𝐶 𝐶 = 𝐵)
4 eqid 2610 . . . 4 𝐴 = 𝐴
54, 1refbas 21123 . . 3 (𝐴Ref𝐵 𝐵 = 𝐴)
63, 5sylan9eqr 2666 . 2 ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐶 = 𝐴)
7 refssex 21124 . . . . . 6 ((𝐴Ref𝐵𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
87ex 449 . . . . 5 (𝐴Ref𝐵 → (𝑥𝐴 → ∃𝑦𝐵 𝑥𝑦))
98adantr 480 . . . 4 ((𝐴Ref𝐵𝐵Ref𝐶) → (𝑥𝐴 → ∃𝑦𝐵 𝑥𝑦))
10 refssex 21124 . . . . . . 7 ((𝐵Ref𝐶𝑦𝐵) → ∃𝑧𝐶 𝑦𝑧)
1110ad2ant2lr 780 . . . . . 6 (((𝐴Ref𝐵𝐵Ref𝐶) ∧ (𝑦𝐵𝑥𝑦)) → ∃𝑧𝐶 𝑦𝑧)
12 sstr2 3575 . . . . . . . 8 (𝑥𝑦 → (𝑦𝑧𝑥𝑧))
1312reximdv 2999 . . . . . . 7 (𝑥𝑦 → (∃𝑧𝐶 𝑦𝑧 → ∃𝑧𝐶 𝑥𝑧))
1413ad2antll 761 . . . . . 6 (((𝐴Ref𝐵𝐵Ref𝐶) ∧ (𝑦𝐵𝑥𝑦)) → (∃𝑧𝐶 𝑦𝑧 → ∃𝑧𝐶 𝑥𝑧))
1511, 14mpd 15 . . . . 5 (((𝐴Ref𝐵𝐵Ref𝐶) ∧ (𝑦𝐵𝑥𝑦)) → ∃𝑧𝐶 𝑥𝑧)
1615rexlimdvaa 3014 . . . 4 ((𝐴Ref𝐵𝐵Ref𝐶) → (∃𝑦𝐵 𝑥𝑦 → ∃𝑧𝐶 𝑥𝑧))
179, 16syld 46 . . 3 ((𝐴Ref𝐵𝐵Ref𝐶) → (𝑥𝐴 → ∃𝑧𝐶 𝑥𝑧))
1817ralrimiv 2948 . 2 ((𝐴Ref𝐵𝐵Ref𝐶) → ∀𝑥𝐴𝑧𝐶 𝑥𝑧)
19 refrel 21121 . . . . 5 Rel Ref
2019brrelexi 5082 . . . 4 (𝐴Ref𝐵𝐴 ∈ V)
2120adantr 480 . . 3 ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴 ∈ V)
224, 2isref 21122 . . 3 (𝐴 ∈ V → (𝐴Ref𝐶 ↔ ( 𝐶 = 𝐴 ∧ ∀𝑥𝐴𝑧𝐶 𝑥𝑧)))
2321, 22syl 17 . 2 ((𝐴Ref𝐵𝐵Ref𝐶) → (𝐴Ref𝐶 ↔ ( 𝐶 = 𝐴 ∧ ∀𝑥𝐴𝑧𝐶 𝑥𝑧)))
246, 18, 23mpbir2and 959 1 ((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴Ref𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  Refcref 21115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-ref 21118 This theorem is referenced by:  refssfne  31523
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