Theorem brtxp 32324

Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 32322, 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 32298	. . 3 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2	1	breqi 4793	. 2 ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3		brin 4839	. 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 5061	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
6	4, 5	brco 5430	. . . 4 ⊢ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7		ancom 448	. . . . . 6 ⊢ ((𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐴𝑦))
8		vex 3354	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8, 5	brcnv 5442	. . . . . . . 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 5305	. . . . . . . . 9 ⊢ ⟨𝑌, 𝑍⟩ ∈ (V × V)
13	8	brres 5542	. . . . . . . . 9 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩1st 𝑦 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
14	12, 13	mpbiran2 689	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
15	10, 11	br1steq 32008	. . . . . . . 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 610	. . . . . 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 1924	. . . 4 ⊢ (∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
20		breq2 4791	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋𝐴𝑦 ↔ 𝑋𝐴𝑌))
21	10, 20	ceqsexv 3394	. . . 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 5430	. . . 4 ⊢ (𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
24		ancom 448	. . . . . 6 ⊢ ((𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐵𝑧))
25		vex 3354	. . . . . . . . 9 ⊢ 𝑧 ∈ V
26	25, 5	brcnv 5442	. . . . . . . 8 ⊢ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
27	25	brres 5542	. . . . . . . . 9 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩2nd 𝑧 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
28	12, 27	mpbiran2 689	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
29	10, 11	br2ndeq 32009	. . . . . . . 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 610	. . . . . 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 1924	. . . 4 ⊢ (∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
34		breq2 4791	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝐵𝑧 ↔ 𝑋𝐵𝑍))
35	11, 34	ceqsexv 3394	. . . 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 612	. 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 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  Vcvv 3351   ∩ cin 3722  ⟨cop 4323   class class class wbr 4787   × cxp 5248  ^◡ccnv 5249   ↾ cres 5252   ∘ ccom 5254  1^st c1st 7317  2^nd c2nd 7318   ⊗ ctxp 32274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fo 6036  df-fv 6038  df-1st 7319  df-2nd 7320  df-txp 32298

This theorem is referenced by:  brtxp2  32325  pprodss4v  32328  brpprod  32329  brsset  32333  brtxpsd  32338  elfuns  32359