Theorem iserd 5943

Description: A symmetric, transitive relationship is an equivalence relationship. (Contributed by SF, 22-Feb-2015.)

Hypotheses

Ref	Expression
iserd.1	⊢ (φ → R ∈ V)
iserd.2	⊢ (φ → A ∈ W)
iserd.3	⊢ ((φ ∧ (x ∈ A ∧ y ∈ A) ∧ xRy) → yRx)
iserd.4	⊢ ((φ ∧ (x ∈ A ∧ y ∈ A ∧ z ∈ A) ∧ (xRy ∧ yRz)) → xRz)

Assertion

Ref	Expression
iserd	⊢ (φ → R Er 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 iserd

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

Step	Hyp	Ref	Expression
1		iserd.3	. . . . 5 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A) ∧ xRy) → yRx)
2	1	3expia 1153	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → (xRy → yRx))
3	2	ralrimivva 2707	. . 3 ⊢ (φ → ∀x ∈ A ∀y ∈ A (xRy → yRx))
4		iserd.1	. . . 4 ⊢ (φ → R ∈ V)
5		iserd.2	. . . 4 ⊢ (φ → A ∈ W)
6		breq 4642	. . . . . . 7 ⊢ (r = R → (xry ↔ xRy))
7		breq 4642	. . . . . . 7 ⊢ (r = R → (yrx ↔ yRx))
8	6, 7	imbi12d 311	. . . . . 6 ⊢ (r = R → ((xry → yrx) ↔ (xRy → yRx)))
9	8	2ralbidv 2657	. . . . 5 ⊢ (r = R → (∀x ∈ a ∀y ∈ a (xry → yrx) ↔ ∀x ∈ a ∀y ∈ a (xRy → yRx)))
10		raleq 2808	. . . . . 6 ⊢ (a = A → (∀y ∈ a (xRy → yRx) ↔ ∀y ∈ A (xRy → yRx)))
11	10	raleqbi1dv 2816	. . . . 5 ⊢ (a = A → (∀x ∈ a ∀y ∈ a (xRy → yRx) ↔ ∀x ∈ A ∀y ∈ A (xRy → yRx)))
12		df-sym 5909	. . . . 5 ⊢ Sym = {⟨r, a⟩ ∣ ∀x ∈ a ∀y ∈ a (xry → yrx)}
13	9, 11, 12	brabg 4707	. . . 4 ⊢ ((R ∈ V ∧ A ∈ W) → (R Sym A ↔ ∀x ∈ A ∀y ∈ A (xRy → yRx)))
14	4, 5, 13	syl2anc 642	. . 3 ⊢ (φ → (R Sym A ↔ ∀x ∈ A ∀y ∈ A (xRy → yRx)))
15	3, 14	mpbird 223	. 2 ⊢ (φ → R Sym A)
16		iserd.4	. . 3 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A ∧ z ∈ A) ∧ (xRy ∧ yRz)) → xRz)
17	4, 5, 16	trrd 5926	. 2 ⊢ (φ → R Trans A)
18		ersymtr 5933	. 2 ⊢ (R Er A ↔ (R Sym A ∧ R Trans A))
19	15, 17, 18	sylanbrc 645	1 ⊢ (φ → R Er 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   Sym csym 5898   Er cer 5899

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  df-sym 5909  df-er 5910

This theorem is referenced by:  ider  5944  eqer  5962  ener  6040