Theorem reftr 23743

Description: Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Assertion

Ref	Expression
reftr	⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴Ref𝐶)

Proof of Theorem reftr

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵
2		eqid 2760	. . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶
3	1, 2	refbas 23739	. . 3 ⊢ (𝐵Ref𝐶 → ∪ 𝐶 = ∪ 𝐵)
4		eqid 2760	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
5	4, 1	refbas 23739	. . 3 ⊢ (𝐴Ref𝐵 → ∪ 𝐵 = ∪ 𝐴)
6	3, 5	sylan9eqr 2817	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∪ 𝐶 = ∪ 𝐴)
7		refssex 23740	. . . . . 6 ⊢ ((𝐴Ref𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
8	7	ex 418	. . . . 5 ⊢ (𝐴Ref𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
9	8	adantr 486	. . . 4 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
10		refssex 23740	. . . . . . 7 ⊢ ((𝐵Ref𝐶 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧)
11	10	ad2ant2lr 761	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑦)) → ∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧)
12		sstr2 3938	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑧 → 𝑥 ⊆ 𝑧))
13	12	reximdv 3177	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑦 → (∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
14	13	ad2antll 742	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑦)) → (∃𝑧 ∈ 𝐶 𝑦 ⊆ 𝑧 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
15	11, 14	mpd 16	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ⊆ 𝑦)) → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)
16	15	rexlimdvaa 3164	. . . 4 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
17	9, 16	syld 48	. . 3 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (𝑥 ∈ 𝐴 → ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧))
18	17	ralrimiv 3153	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)
19		refrel 23737	. . . . 5 ⊢ Rel Ref
20	19	brrelex1i 5711	. . . 4 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
21	20	adantr 486	. . 3 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴 ∈ V)
22	4, 2	isref 23738	. . 3 ⊢ (𝐴 ∈ V → (𝐴Ref𝐶 ↔ (∪ 𝐶 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)))
23	21, 22	syl 18	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → (𝐴Ref𝐶 ↔ (∪ 𝐶 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 ⊆ 𝑧)))
24	6, 18, 23	mpbir2and 726	1 ⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴Ref𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  ∪ cuni 4867   class class class wbr 5103  Refcref 23731

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-ref 23734

This theorem is used by:  refssfne  36980