Theorem brtxp 36113

Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 36111, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

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 36087	. . 3 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2	1	breqi 5085	. 2 ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3		brin 5131	. 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 5410	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
6	4, 5	brco 5819	. . . 4 ⊢ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7		vex 3436	. . . . . . . 8 ⊢ 𝑦 ∈ V
8	7, 5	brcnv 5831	. . . . . . 7 ⊢ (𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦)
9		brtxp.2	. . . . . . . . 9 ⊢ 𝑌 ∈ V
10		brtxp.3	. . . . . . . . 9 ⊢ 𝑍 ∈ V
11	9, 10	opelvv 5665	. . . . . . . 8 ⊢ ⟨𝑌, 𝑍⟩ ∈ (V × V)
12	7	brresi 5947	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩1st 𝑦))
13	11, 12	mpbiran 715	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
14	9, 10	br1steq 36006	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩1st 𝑦 ↔ 𝑦 = 𝑌)
15	8, 13, 14	3bitri 298	. . . . . 6 ⊢ (𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑦 = 𝑌)
16	15	anbi1ci 632	. . . . 5 ⊢ ((𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
17	16	exbii 1855	. . . 4 ⊢ (∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
18		breq2 5083	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋𝐴𝑦 ↔ 𝑋𝐴𝑌))
19	9, 18	ceqsexv 3481	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑌 ∧ 𝑋𝐴𝑦) ↔ 𝑋𝐴𝑌)
20	6, 17, 19	3bitri 298	. . 3 ⊢ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐴𝑌)
21	4, 5	brco 5819	. . . 4 ⊢ (𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
22		vex 3436	. . . . . . . 8 ⊢ 𝑧 ∈ V
23	22, 5	brcnv 5831	. . . . . . 7 ⊢ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
24	22	brresi 5947	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩2nd 𝑧))
25	11, 24	mpbiran 715	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
26	9, 10	br2ndeq 36007	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩2nd 𝑧 ↔ 𝑧 = 𝑍)
27	23, 25, 26	3bitri 298	. . . . . 6 ⊢ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑧 = 𝑍)
28	27	anbi1ci 632	. . . . 5 ⊢ ((𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
29	28	exbii 1855	. . . 4 ⊢ (∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
30		breq2 5083	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝐵𝑧 ↔ 𝑋𝐵𝑍))
31	10, 30	ceqsexv 3481	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑍 ∧ 𝑋𝐵𝑧) ↔ 𝑋𝐵𝑍)
32	21, 29, 31	3bitri 298	. . 3 ⊢ (𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐵𝑍)
33	20, 32	anbi12i 634	. 2 ⊢ ((𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ∧ 𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩) ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))
34	2, 3, 33	3bitri 298	1 ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3432   ∩ cin 3889  ⟨cop 4568   class class class wbr 5079   × cxp 5623  ^◡ccnv 5624   ↾ cres 5627   ∘ ccom 5629  1^st c1st 7936  2^nd c2nd 7937   ⊗ ctxp 36063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7938  df-2nd 7939  df-txp 36087

This theorem is referenced by:  brtxp2  36114  pprodss4v  36117  brpprod  36118  brsset  36122  brtxpsd  36127  elfuns  36148