Theorem br1stg 4731

Description: The binary relationship over the 1^st function. (Contributed by SF, 5-Jan-2015.)

Assertion

Ref	Expression
br1stg	⊢ ((A ∈ V ∧ B ∈ W) → (⟨A, B⟩1st C ↔ A = C))

Proof of Theorem br1stg

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

Step	Hyp	Ref	Expression
1		opeq1 4579	. . . 4 ⊢ (z = A → ⟨z, w⟩ = ⟨A, w⟩)
2	1	breq1d 4650	. . 3 ⊢ (z = A → (⟨z, w⟩1st C ↔ ⟨A, w⟩1st C))
3		eqeq1 2359	. . 3 ⊢ (z = A → (z = C ↔ A = C))
4	2, 3	bibi12d 312	. 2 ⊢ (z = A → ((⟨z, w⟩1st C ↔ z = C) ↔ (⟨A, w⟩1st C ↔ A = C)))
5		opeq2 4580	. . . 4 ⊢ (w = B → ⟨A, w⟩ = ⟨A, B⟩)
6	5	breq1d 4650	. . 3 ⊢ (w = B → (⟨A, w⟩1st C ↔ ⟨A, B⟩1st C))
7	6	bibi1d 310	. 2 ⊢ (w = B → ((⟨A, w⟩1st C ↔ A = C) ↔ (⟨A, B⟩1st C ↔ A = C)))
8		df-br 4641	. . 3 ⊢ (⟨z, w⟩1st C ↔ ⟨⟨z, w⟩, C⟩ ∈ 1st )
9		el1st 4730	. . 3 ⊢ (⟨⟨z, w⟩, C⟩ ∈ 1st ↔ ∃x∃y⟨⟨z, w⟩, C⟩ = ⟨⟨x, y⟩, x⟩)
10		eqcom 2355	. . . . . 6 ⊢ (⟨⟨z, w⟩, C⟩ = ⟨⟨x, y⟩, x⟩ ↔ ⟨⟨x, y⟩, x⟩ = ⟨⟨z, w⟩, C⟩)
11		opth 4603	. . . . . . 7 ⊢ (⟨⟨x, y⟩, x⟩ = ⟨⟨z, w⟩, C⟩ ↔ (⟨x, y⟩ = ⟨z, w⟩ ∧ x = C))
12		opth 4603	. . . . . . . . 9 ⊢ (⟨x, y⟩ = ⟨z, w⟩ ↔ (x = z ∧ y = w))
13	12	anbi1i 676	. . . . . . . 8 ⊢ ((⟨x, y⟩ = ⟨z, w⟩ ∧ x = C) ↔ ((x = z ∧ y = w) ∧ x = C))
14		df-3an 936	. . . . . . . 8 ⊢ ((x = z ∧ y = w ∧ x = C) ↔ ((x = z ∧ y = w) ∧ x = C))
15	13, 14	bitr4i 243	. . . . . . 7 ⊢ ((⟨x, y⟩ = ⟨z, w⟩ ∧ x = C) ↔ (x = z ∧ y = w ∧ x = C))
16	11, 15	bitri 240	. . . . . 6 ⊢ (⟨⟨x, y⟩, x⟩ = ⟨⟨z, w⟩, C⟩ ↔ (x = z ∧ y = w ∧ x = C))
17	10, 16	bitri 240	. . . . 5 ⊢ (⟨⟨z, w⟩, C⟩ = ⟨⟨x, y⟩, x⟩ ↔ (x = z ∧ y = w ∧ x = C))
18	17	2exbii 1583	. . . 4 ⊢ (∃x∃y⟨⟨z, w⟩, C⟩ = ⟨⟨x, y⟩, x⟩ ↔ ∃x∃y(x = z ∧ y = w ∧ x = C))
19		vex 2863	. . . . 5 ⊢ z ∈ V
20		vex 2863	. . . . 5 ⊢ w ∈ V
21		eqeq1 2359	. . . . 5 ⊢ (x = z → (x = C ↔ z = C))
22		biidd 228	. . . . 5 ⊢ (y = w → (z = C ↔ z = C))
23	19, 20, 21, 22	ceqsex2v 2897	. . . 4 ⊢ (∃x∃y(x = z ∧ y = w ∧ x = C) ↔ z = C)
24	18, 23	bitri 240	. . 3 ⊢ (∃x∃y⟨⟨z, w⟩, C⟩ = ⟨⟨x, y⟩, x⟩ ↔ z = C)
25	8, 9, 24	3bitri 262	. 2 ⊢ (⟨z, w⟩1st C ↔ z = C)
26	4, 7, 25	vtocl2g 2919	1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟨A, B⟩1st C ↔ A = C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718

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-1st 4724

This theorem is referenced by: (None)