Theorem xpassenlem 6057

Description: Lemma for xpassen 6058. Compute a projection. (Contributed by Scott Fenton, 19-Apr-2021.)

Assertion

Ref	Expression
xpassenlem	⊢ (y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ↔ ( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x))

Proof of Theorem xpassenlem

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq 4620	. . . . 5 ⊢ y = ⟨ Proj1 y, Proj2 y⟩
2	1	breq1i 4647	. . . 4 ⊢ (y(1st ∘ 1st ) Proj1 x ↔ ⟨ Proj1 y, Proj2 y⟩(1st ∘ 1st ) Proj1 x)
3		brco 4884	. . . 4 ⊢ (⟨ Proj1 y, Proj2 y⟩(1st ∘ 1st ) Proj1 x ↔ ∃t(⟨ Proj1 y, Proj2 y⟩1st t ∧ t1st Proj1 x))
4		vex 2863	. . . . . . . . . 10 ⊢ y ∈ V
5	4	proj1ex 4594	. . . . . . . . 9 ⊢ Proj1 y ∈ V
6	4	proj2ex 4595	. . . . . . . . 9 ⊢ Proj2 y ∈ V
7	5, 6	opbr1st 5502	. . . . . . . 8 ⊢ (⟨ Proj1 y, Proj2 y⟩1st t ↔ Proj1 y = t)
8		eqcom 2355	. . . . . . . 8 ⊢ ( Proj1 y = t ↔ t = Proj1 y)
9	7, 8	bitri 240	. . . . . . 7 ⊢ (⟨ Proj1 y, Proj2 y⟩1st t ↔ t = Proj1 y)
10	9	anbi1i 676	. . . . . 6 ⊢ ((⟨ Proj1 y, Proj2 y⟩1st t ∧ t1st Proj1 x) ↔ (t = Proj1 y ∧ t1st Proj1 x))
11	10	exbii 1582	. . . . 5 ⊢ (∃t(⟨ Proj1 y, Proj2 y⟩1st t ∧ t1st Proj1 x) ↔ ∃t(t = Proj1 y ∧ t1st Proj1 x))
12		breq1 4643	. . . . . . 7 ⊢ (t = Proj1 y → (t1st Proj1 x ↔ Proj1 y1st Proj1 x))
13		opeq 4620	. . . . . . . . 9 ⊢ Proj1 y = ⟨ Proj1 Proj1 y, Proj2 Proj1 y⟩
14	13	breq1i 4647	. . . . . . . 8 ⊢ ( Proj1 y1st Proj1 x ↔ ⟨ Proj1 Proj1 y, Proj2 Proj1 y⟩1st Proj1 x)
15	5	proj1ex 4594	. . . . . . . . 9 ⊢ Proj1 Proj1 y ∈ V
16	5	proj2ex 4595	. . . . . . . . 9 ⊢ Proj2 Proj1 y ∈ V
17	15, 16	opbr1st 5502	. . . . . . . 8 ⊢ (⟨ Proj1 Proj1 y, Proj2 Proj1 y⟩1st Proj1 x ↔ Proj1 Proj1 y = Proj1 x)
18	14, 17	bitri 240	. . . . . . 7 ⊢ ( Proj1 y1st Proj1 x ↔ Proj1 Proj1 y = Proj1 x)
19	12, 18	syl6bb 252	. . . . . 6 ⊢ (t = Proj1 y → (t1st Proj1 x ↔ Proj1 Proj1 y = Proj1 x))
20	5, 19	ceqsexv 2895	. . . . 5 ⊢ (∃t(t = Proj1 y ∧ t1st Proj1 x) ↔ Proj1 Proj1 y = Proj1 x)
21	11, 20	bitri 240	. . . 4 ⊢ (∃t(⟨ Proj1 y, Proj2 y⟩1st t ∧ t1st Proj1 x) ↔ Proj1 Proj1 y = Proj1 x)
22	2, 3, 21	3bitri 262	. . 3 ⊢ (y(1st ∘ 1st ) Proj1 x ↔ Proj1 Proj1 y = Proj1 x)
23		opeq 4620	. . . . 5 ⊢ Proj2 x = ⟨ Proj1 Proj2 x, Proj2 Proj2 x⟩
24	23	breq2i 4648	. . . 4 ⊢ (y((2nd ∘ 1st ) ⊗ 2nd ) Proj2 x ↔ y((2nd ∘ 1st ) ⊗ 2nd )⟨ Proj1 Proj2 x, Proj2 Proj2 x⟩)
25		trtxp 5782	. . . 4 ⊢ (y((2nd ∘ 1st ) ⊗ 2nd )⟨ Proj1 Proj2 x, Proj2 Proj2 x⟩ ↔ (y(2nd ∘ 1st ) Proj1 Proj2 x ∧ y2nd Proj2 Proj2 x))
26	1	breq1i 4647	. . . . . 6 ⊢ (y(2nd ∘ 1st ) Proj1 Proj2 x ↔ ⟨ Proj1 y, Proj2 y⟩(2nd ∘ 1st ) Proj1 Proj2 x)
27		brco 4884	. . . . . 6 ⊢ (⟨ Proj1 y, Proj2 y⟩(2nd ∘ 1st ) Proj1 Proj2 x ↔ ∃t(⟨ Proj1 y, Proj2 y⟩1st t ∧ t2nd Proj1 Proj2 x))
28	9	anbi1i 676	. . . . . . . 8 ⊢ ((⟨ Proj1 y, Proj2 y⟩1st t ∧ t2nd Proj1 Proj2 x) ↔ (t = Proj1 y ∧ t2nd Proj1 Proj2 x))
29	28	exbii 1582	. . . . . . 7 ⊢ (∃t(⟨ Proj1 y, Proj2 y⟩1st t ∧ t2nd Proj1 Proj2 x) ↔ ∃t(t = Proj1 y ∧ t2nd Proj1 Proj2 x))
30		breq1 4643	. . . . . . . . 9 ⊢ (t = Proj1 y → (t2nd Proj1 Proj2 x ↔ Proj1 y2nd Proj1 Proj2 x))
31	13	breq1i 4647	. . . . . . . . . 10 ⊢ ( Proj1 y2nd Proj1 Proj2 x ↔ ⟨ Proj1 Proj1 y, Proj2 Proj1 y⟩2nd Proj1 Proj2 x)
32	15, 16	opbr2nd 5503	. . . . . . . . . 10 ⊢ (⟨ Proj1 Proj1 y, Proj2 Proj1 y⟩2nd Proj1 Proj2 x ↔ Proj2 Proj1 y = Proj1 Proj2 x)
33	31, 32	bitri 240	. . . . . . . . 9 ⊢ ( Proj1 y2nd Proj1 Proj2 x ↔ Proj2 Proj1 y = Proj1 Proj2 x)
34	30, 33	syl6bb 252	. . . . . . . 8 ⊢ (t = Proj1 y → (t2nd Proj1 Proj2 x ↔ Proj2 Proj1 y = Proj1 Proj2 x))
35	5, 34	ceqsexv 2895	. . . . . . 7 ⊢ (∃t(t = Proj1 y ∧ t2nd Proj1 Proj2 x) ↔ Proj2 Proj1 y = Proj1 Proj2 x)
36	29, 35	bitri 240	. . . . . 6 ⊢ (∃t(⟨ Proj1 y, Proj2 y⟩1st t ∧ t2nd Proj1 Proj2 x) ↔ Proj2 Proj1 y = Proj1 Proj2 x)
37	26, 27, 36	3bitri 262	. . . . 5 ⊢ (y(2nd ∘ 1st ) Proj1 Proj2 x ↔ Proj2 Proj1 y = Proj1 Proj2 x)
38	1	breq1i 4647	. . . . . 6 ⊢ (y2nd Proj2 Proj2 x ↔ ⟨ Proj1 y, Proj2 y⟩2nd Proj2 Proj2 x)
39	5, 6	opbr2nd 5503	. . . . . 6 ⊢ (⟨ Proj1 y, Proj2 y⟩2nd Proj2 Proj2 x ↔ Proj2 y = Proj2 Proj2 x)
40	38, 39	bitri 240	. . . . 5 ⊢ (y2nd Proj2 Proj2 x ↔ Proj2 y = Proj2 Proj2 x)
41	37, 40	anbi12i 678	. . . 4 ⊢ ((y(2nd ∘ 1st ) Proj1 Proj2 x ∧ y2nd Proj2 Proj2 x) ↔ ( Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x))
42	24, 25, 41	3bitri 262	. . 3 ⊢ (y((2nd ∘ 1st ) ⊗ 2nd ) Proj2 x ↔ ( Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x))
43	22, 42	anbi12i 678	. 2 ⊢ ((y(1st ∘ 1st ) Proj1 x ∧ y((2nd ∘ 1st ) ⊗ 2nd ) Proj2 x) ↔ ( Proj1 Proj1 y = Proj1 x ∧ ( Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x)))
44		opeq 4620	. . . 4 ⊢ x = ⟨ Proj1 x, Proj2 x⟩
45	44	breq2i 4648	. . 3 ⊢ (y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ↔ y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨ Proj1 x, Proj2 x⟩)
46		trtxp 5782	. . 3 ⊢ (y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨ Proj1 x, Proj2 x⟩ ↔ (y(1st ∘ 1st ) Proj1 x ∧ y((2nd ∘ 1st ) ⊗ 2nd ) Proj2 x))
47	45, 46	bitri 240	. 2 ⊢ (y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ↔ (y(1st ∘ 1st ) Proj1 x ∧ y((2nd ∘ 1st ) ⊗ 2nd ) Proj2 x))
48		3anass 938	. 2 ⊢ (( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) ↔ ( Proj1 Proj1 y = Proj1 x ∧ ( Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x)))
49	43, 47, 48	3bitr4i 268	1 ⊢ (y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ↔ ( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642  ⟨cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722  2^nd c2nd 4784   ⊗ ctxp 5736

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  df-co 4727  df-cnv 4786  df-2nd 4798  df-txp 5737

This theorem is referenced by:  xpassen  6058