Theorem brtxp 31987

Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 31985, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)

Hypotheses

Ref	Expression
brtxp.1	⊢ 𝑋 ∈ V
brtxp.2	⊢ 𝑌 ∈ V
brtxp.3	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
brtxp	⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))

Proof of Theorem brtxp

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

Step	Hyp	Ref	Expression
1		df-txp 31961	. . 3 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2	1	breqi 4659	. 2 ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3		brin 4704	. 2 ⊢ (𝑋((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩ ↔ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ∧ 𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩))
4		brtxp.1	. . . . 5 ⊢ 𝑋 ∈ V
5		opex 4932	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
6	4, 5	brco 5292	. . . 4 ⊢ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7		ancom 466	. . . . . 6 ⊢ ((𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐴𝑦))
8		vex 3203	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8, 5	brcnv 5305	. . . . . . . 8 ⊢ (𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦)
10		brtxp.2	. . . . . . . . . 10 ⊢ 𝑌 ∈ V
11		brtxp.3	. . . . . . . . . 10 ⊢ 𝑍 ∈ V
12	10, 11	opelvv 5166	. . . . . . . . 9 ⊢ ⟨𝑌, 𝑍⟩ ∈ (V × V)
13	8	brres 5402	. . . . . . . . 9 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩1st 𝑦 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
14	12, 13	mpbiran2 954	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
15	10, 11	br1steq 31670	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩1st 𝑦 ↔ 𝑦 = 𝑌)
16	9, 14, 15	3bitri 286	. . . . . . 7 ⊢ (𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑦 = 𝑌)
17	16	anbi1i 731	. . . . . 6 ⊢ ((𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐴𝑦) ↔ (𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
18	7, 17	bitri 264	. . . . 5 ⊢ ((𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
19	18	exbii 1774	. . . 4 ⊢ (∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
20		breq2 4657	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋𝐴𝑦 ↔ 𝑋𝐴𝑌))
21	10, 20	ceqsexv 3242	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑌 ∧ 𝑋𝐴𝑦) ↔ 𝑋𝐴𝑌)
22	6, 19, 21	3bitri 286	. . 3 ⊢ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐴𝑌)
23	4, 5	brco 5292	. . . 4 ⊢ (𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
24		ancom 466	. . . . . 6 ⊢ ((𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐵𝑧))
25		vex 3203	. . . . . . . . 9 ⊢ 𝑧 ∈ V
26	25, 5	brcnv 5305	. . . . . . . 8 ⊢ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
27	25	brres 5402	. . . . . . . . 9 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩2nd 𝑧 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
28	12, 27	mpbiran2 954	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
29	10, 11	br2ndeq 31671	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩2nd 𝑧 ↔ 𝑧 = 𝑍)
30	26, 28, 29	3bitri 286	. . . . . . 7 ⊢ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑧 = 𝑍)
31	30	anbi1i 731	. . . . . 6 ⊢ ((𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐵𝑧) ↔ (𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
32	24, 31	bitri 264	. . . . 5 ⊢ ((𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
33	32	exbii 1774	. . . 4 ⊢ (∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
34		breq2 4657	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝐵𝑧 ↔ 𝑋𝐵𝑍))
35	11, 34	ceqsexv 3242	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑍 ∧ 𝑋𝐵𝑧) ↔ 𝑋𝐵𝑍)
36	23, 33, 35	3bitri 286	. . 3 ⊢ (𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐵𝑍)
37	22, 36	anbi12i 733	. 2 ⊢ ((𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ∧ 𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩) ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))
38	2, 3, 37	3bitri 286	1 ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113   ↾ cres 5116   ∘ ccom 5118  1^st c1st 7166  2^nd c2nd 7167   ⊗ ctxp 31937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961

This theorem is referenced by:  brtxp2  31988  pprodss4v  31991  brpprod  31992  brsset  31996  brtxpsd  32001  elfuns  32022