Theorem brtxp2 36080

Description: The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)

Hypothesis

Ref	Expression
brtxp2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
brtxp2	⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Proof of Theorem brtxp2

Step	Hyp	Ref	Expression
1		txpss3v 36077	. . . . . . 7 ⊢ (𝑅 ⊗ 𝑆) ⊆ (V × (V × V))
2	1	brel 5690	. . . . . 6 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V)))
3	2	simprd 495	. . . . 5 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → 𝐵 ∈ (V × V))
4		elvv 5700	. . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
5	3, 4	sylib 218	. . . 4 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
6	5	pm4.71ri 560	. . 3 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵))
7		19.41vv 1952	. . 3 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵))
8	6, 7	bitr4i 278	. 2 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵))
9		breq2 5090	. . . 4 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩))
10	9	pm5.32i 574	. . 3 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩))
11	10	2exbii 1851	. 2 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩))
12		brtxp2.1	. . . . . 6 ⊢ 𝐴 ∈ V
13		vex 3434	. . . . . 6 ⊢ 𝑥 ∈ V
14		vex 3434	. . . . . 6 ⊢ 𝑦 ∈ V
15	12, 13, 14	brtxp 36079	. . . . 5 ⊢ (𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
16	15	anbi2i 624	. . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
17		3anass 1095	. . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
18	16, 17	bitr4i 278	. . 3 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
19	18	2exbii 1851	. 2 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
20	8, 11, 19	3bitri 297	1 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   class class class wbr 5086   × cxp 5623   ⊗ ctxp 36029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  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 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7936  df-2nd 7937  df-txp 36053

This theorem is referenced by:  lemsuccf  36140  brrestrict  36150