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Theorem pprodexg 5837
 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
StepHypRef Expression
1 df-pprod 5738 . 2 PProd (A, B) = ((A 1st ) ⊗ (B 2nd ))
2 1stex 4739 . . . 4 1st V
3 coexg 4749 . . . 4 ((A V 1st V) → (A 1st ) V)
42, 3mpan2 652 . . 3 (A V → (A 1st ) V)
5 2ndex 5112 . . . 4 2nd V
6 coexg 4749 . . . 4 ((B W 2nd V) → (B 2nd ) V)
75, 6mpan2 652 . . 3 (B W → (B 2nd ) V)
8 txpexg 5784 . . 3 (((A 1st ) V (B 2nd ) V) → ((A 1st ) ⊗ (B 2nd )) V)
94, 7, 8syl2an 463 . 2 ((A V B W) → ((A 1st ) ⊗ (B 2nd )) V)
101, 9syl5eqel 2437 1 ((A V B W) → PProd (A, B) V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2859  1st c1st 4717   ∘ ccom 4721  2nd c2nd 4783   ⊗ ctxp 5735   PProd cpprod 5737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-co 4726  df-ima 4727  df-cnv 4785  df-2nd 4797  df-txp 5736  df-pprod 5738 This theorem is referenced by:  pprodex  5838  frecexg  6312  dmfrec  6316  fnfreclem2  6318  fnfreclem3  6319  frec0  6321  frecsuc  6322
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