Theorem eqer 5961

Description: Equivalence relation involving equality of dependent classes A(x) and B(y). (Contributed by set.mm contributors, 17-Mar-2008.)

Hypotheses

Ref	Expression
eqer.1	⊢ (x = y → A = B)
eqer.2	⊢ R = {⟨x, y⟩ ∣ A = B}
eqer.3	⊢ R ∈ V

Assertion

Ref	Expression
eqer	⊢ R Er V

Distinct variable groups:   y,A   x,B   x,y

Allowed substitution hints:   A(x)   B(y)   R(x,y)

Proof of Theorem eqer

Dummy variables v w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqer.3	. . . 4 ⊢ R ∈ V
2	1	a1i 10	. . 3 ⊢ ( ⊤ → R ∈ V)
3		vvex 4109	. . . 4 ⊢ V ∈ V
4	3	a1i 10	. . 3 ⊢ ( ⊤ → V ∈ V)
5		id 19	. . . . . 6 ⊢ ([z / x]A = [w / x]A → [z / x]A = [w / x]A)
6	5	eqcomd 2358	. . . . 5 ⊢ ([z / x]A = [w / x]A → [w / x]A = [z / x]A)
7		eqer.1	. . . . . 6 ⊢ (x = y → A = B)
8		eqer.2	. . . . . 6 ⊢ R = {⟨x, y⟩ ∣ A = B}
9	7, 8	eqerlem 5960	. . . . 5 ⊢ (zRw ↔ [z / x]A = [w / x]A)
10	7, 8	eqerlem 5960	. . . . 5 ⊢ (wRz ↔ [w / x]A = [z / x]A)
11	6, 9, 10	3imtr4i 257	. . . 4 ⊢ (zRw → wRz)
12	11	3ad2ant3 978	. . 3 ⊢ (( ⊤ ∧ (z ∈ V ∧ w ∈ V) ∧ zRw) → wRz)
13		eqtr 2370	. . . . 5 ⊢ (([z / x]A = [w / x]A ∧ [w / x]A = [v / x]A) → [z / x]A = [v / x]A)
14	7, 8	eqerlem 5960	. . . . . 6 ⊢ (wRv ↔ [w / x]A = [v / x]A)
15	9, 14	anbi12i 678	. . . . 5 ⊢ ((zRw ∧ wRv) ↔ ([z / x]A = [w / x]A ∧ [w / x]A = [v / x]A))
16	7, 8	eqerlem 5960	. . . . 5 ⊢ (zRv ↔ [z / x]A = [v / x]A)
17	13, 15, 16	3imtr4i 257	. . . 4 ⊢ ((zRw ∧ wRv) → zRv)
18	17	3ad2ant3 978	. . 3 ⊢ (( ⊤ ∧ (z ∈ V ∧ w ∈ V ∧ v ∈ V) ∧ (zRw ∧ wRv)) → zRv)
19	2, 4, 12, 18	iserd 5942	. 2 ⊢ ( ⊤ → R Er V)
20	19	trud 1323	1 ⊢ R Er V

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   ⊤ wtru 1316   = wceq 1642   ∈ wcel 1710  Vcvv 2859  [csb 3136  {copab 4622   class class class wbr 4639   Er cer 5898

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-trans 5899  df-sym 5908  df-er 5909

This theorem is referenced by: (None)