Theorem trrd 5926

Description: Deduce transitivity from its properties. (Contributed by SF, 22-Feb-2015.)

Hypotheses

Ref	Expression
trrd.1	⊢ (φ → R ∈ V)
trrd.2	⊢ (φ → A ∈ W)
trrd.3	⊢ ((φ ∧ (x ∈ A ∧ y ∈ A ∧ z ∈ A) ∧ (xRy ∧ yRz)) → xRz)

Assertion

Ref	Expression
trrd	⊢ (φ → R Trans A)

Distinct variable groups:   x,A,y,z   φ,x,y,z   x,R,y,z

Allowed substitution hints:   V(x,y,z)   W(x,y,z)

Proof of Theorem trrd

Dummy variables a r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 936	. . . . . 6 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) ↔ ((x ∈ A ∧ y ∈ A) ∧ z ∈ A))
2		trrd.3	. . . . . . 7 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A ∧ z ∈ A) ∧ (xRy ∧ yRz)) → xRz)
3	2	3exp 1150	. . . . . 6 ⊢ (φ → ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → ((xRy ∧ yRz) → xRz)))
4	1, 3	syl5bir 209	. . . . 5 ⊢ (φ → (((x ∈ A ∧ y ∈ A) ∧ z ∈ A) → ((xRy ∧ yRz) → xRz)))
5	4	exp3a 425	. . . 4 ⊢ (φ → ((x ∈ A ∧ y ∈ A) → (z ∈ A → ((xRy ∧ yRz) → xRz))))
6	5	ralrimdv 2704	. . 3 ⊢ (φ → ((x ∈ A ∧ y ∈ A) → ∀z ∈ A ((xRy ∧ yRz) → xRz)))
7	6	ralrimivv 2706	. 2 ⊢ (φ → ∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz))
8		trrd.1	. . 3 ⊢ (φ → R ∈ V)
9		trrd.2	. . 3 ⊢ (φ → A ∈ W)
10		breq 4642	. . . . . . . 8 ⊢ (r = R → (xry ↔ xRy))
11		breq 4642	. . . . . . . 8 ⊢ (r = R → (yrz ↔ yRz))
12	10, 11	anbi12d 691	. . . . . . 7 ⊢ (r = R → ((xry ∧ yrz) ↔ (xRy ∧ yRz)))
13		breq 4642	. . . . . . 7 ⊢ (r = R → (xrz ↔ xRz))
14	12, 13	imbi12d 311	. . . . . 6 ⊢ (r = R → (((xry ∧ yrz) → xrz) ↔ ((xRy ∧ yRz) → xRz)))
15	14	ralbidv 2635	. . . . 5 ⊢ (r = R → (∀z ∈ a ((xry ∧ yrz) → xrz) ↔ ∀z ∈ a ((xRy ∧ yRz) → xRz)))
16	15	2ralbidv 2657	. . . 4 ⊢ (r = R → (∀x ∈ a ∀y ∈ a ∀z ∈ a ((xry ∧ yrz) → xrz) ↔ ∀x ∈ a ∀y ∈ a ∀z ∈ a ((xRy ∧ yRz) → xRz)))
17		raleq 2808	. . . . . 6 ⊢ (a = A → (∀z ∈ a ((xRy ∧ yRz) → xRz) ↔ ∀z ∈ A ((xRy ∧ yRz) → xRz)))
18	17	raleqbi1dv 2816	. . . . 5 ⊢ (a = A → (∀y ∈ a ∀z ∈ a ((xRy ∧ yRz) → xRz) ↔ ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz)))
19	18	raleqbi1dv 2816	. . . 4 ⊢ (a = A → (∀x ∈ a ∀y ∈ a ∀z ∈ a ((xRy ∧ yRz) → xRz) ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz)))
20		df-trans 5900	. . . 4 ⊢ Trans = {⟨r, a⟩ ∣ ∀x ∈ a ∀y ∈ a ∀z ∈ a ((xry ∧ yrz) → xrz)}
21	16, 19, 20	brabg 4707	. . 3 ⊢ ((R ∈ V ∧ A ∈ W) → (R Trans A ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz)))
22	8, 9, 21	syl2anc 642	. 2 ⊢ (φ → (R Trans A ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz)))
23	7, 22	mpbird 223	1 ⊢ (φ → R Trans A)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2615   class class class wbr 4640   Trans ctrans 5889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-trans 5900

This theorem is referenced by:  pod  5937  iserd  5943