Theorem pprodexg 5838

Description: The parallel product of two sets is a set. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
pprodexg	⊢ ((A ∈ V ∧ B ∈ W) → PProd (A, B) ∈ V)

Proof of Theorem pprodexg

Step	Hyp	Ref	Expression
1		df-pprod 5739	. 2 ⊢ PProd (A, B) = ((A ∘ 1st ) ⊗ (B ∘ 2nd ))
2		1stex 4740	. . . 4 ⊢ 1st ∈ V
3		coexg 4750	. . . 4 ⊢ ((A ∈ V ∧ 1st ∈ V) → (A ∘ 1st ) ∈ V)
4	2, 3	mpan2 652	. . 3 ⊢ (A ∈ V → (A ∘ 1st ) ∈ V)
5		2ndex 5113	. . . 4 ⊢ 2nd ∈ V
6		coexg 4750	. . . 4 ⊢ ((B ∈ W ∧ 2nd ∈ V) → (B ∘ 2nd ) ∈ V)
7	5, 6	mpan2 652	. . 3 ⊢ (B ∈ W → (B ∘ 2nd ) ∈ V)
8		txpexg 5785	. . 3 ⊢ (((A ∘ 1st ) ∈ V ∧ (B ∘ 2nd ) ∈ V) → ((A ∘ 1st ) ⊗ (B ∘ 2nd )) ∈ V)
9	4, 7, 8	syl2an 463	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → ((A ∘ 1st ) ⊗ (B ∘ 2nd )) ∈ V)
10	1, 9	syl5eqel 2437	1 ⊢ ((A ∈ V ∧ B ∈ W) → PProd (A, B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2860  1^st c1st 4718   ∘ ccom 4722  2^nd c2nd 4784   ⊗ ctxp 5736   PProd cpprod 5738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-co 4727  df-ima 4728  df-cnv 4786  df-2nd 4798  df-txp 5737  df-pprod 5739

This theorem is referenced by:  pprodex  5839  frecexg  6313  dmfrec  6317  fnfreclem2  6319  fnfreclem3  6320  frec0  6322  frecsuc  6323