Theorem brpprod3a 36066

Description: Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brpprod3.1	⊢ 𝑋 ∈ V
brpprod3.2	⊢ 𝑌 ∈ V
brpprod3.3	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
brpprod3a	⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))

Distinct variable groups:   𝑧,𝑤,𝑅   𝑤,𝑆,𝑧   𝑤,𝑋,𝑧   𝑤,𝑌,𝑧   𝑤,𝑍,𝑧

Proof of Theorem brpprod3a

Step	Hyp	Ref	Expression
1		pprodss4v 36064	. . . . . . 7 ⊢ pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V))
2	1	brel 5696	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ 𝑍 ∈ (V × V)))
3	2	simprd 495	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → 𝑍 ∈ (V × V))
4		elvv 5706	. . . . 5 ⊢ (𝑍 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
5	3, 4	sylib 218	. . . 4 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → ∃𝑧∃𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
6	5	pm4.71ri 560	. . 3 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ (∃𝑧∃𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
7		19.41vv 1952	. . 3 ⊢ (∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (∃𝑧∃𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
8	6, 7	bitr4i 278	. 2 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
9		breq2 5089	. . . 4 ⊢ (𝑍 = ⟨𝑧, 𝑤⟩ → (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
10	9	pm5.32i 574	. . 3 ⊢ ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
11	10	2exbii 1851	. 2 ⊢ (∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
12		brpprod3.1	. . . . . 6 ⊢ 𝑋 ∈ V
13		brpprod3.2	. . . . . 6 ⊢ 𝑌 ∈ V
14		vex 3433	. . . . . 6 ⊢ 𝑧 ∈ V
15		vex 3433	. . . . . 6 ⊢ 𝑤 ∈ V
16	12, 13, 14, 15	brpprod 36065	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩ ↔ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))
17	16	anbi2i 624	. . . 4 ⊢ ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)))
18		3anass 1095	. . . 4 ⊢ ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)))
19	17, 18	bitr4i 278	. . 3 ⊢ ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))
20	19	2exbii 1851	. 2 ⊢ (∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))
21	8, 11, 20	3bitri 297	1 ⊢ (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   class class class wbr 5085   × cxp 5629  pprodcpprod 36011

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-1st 7942  df-2nd 7943  df-txp 36034  df-pprod 36035

This theorem is referenced by:  brpprod3b  36067  brapply  36118  dfrdg4  36133