Theorem f1opprod 5845

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

Assertion

Ref	Expression
f1opprod	⊢ ((F:A–1-1-onto→C ∧ G:B–1-1-onto→D) → PProd (F, G):(A × B)–1-1-onto→(C × D))

Proof of Theorem f1opprod

Step	Hyp	Ref	Expression
1		fnpprod 5844	. . . 4 ⊢ ((F Fn A ∧ G Fn B) → PProd (F, G) Fn (A × B))
2		fnpprod 5844	. . . . 5 ⊢ ((◡F Fn C ∧ ◡G Fn D) → PProd (◡F, ◡G) Fn (C × D))
3		cnvpprod 5842	. . . . . 6 ⊢ ◡ PProd (F, G) = PProd (◡F, ◡G)
4	3	fneq1i 5179	. . . . 5 ⊢ (◡ PProd (F, G) Fn (C × D) ↔ PProd (◡F, ◡G) Fn (C × D))
5	2, 4	sylibr 203	. . . 4 ⊢ ((◡F Fn C ∧ ◡G Fn D) → ◡ PProd (F, G) Fn (C × D))
6	1, 5	anim12i 549	. . 3 ⊢ (((F Fn A ∧ G Fn B) ∧ (◡F Fn C ∧ ◡G Fn D)) → ( PProd (F, G) Fn (A × B) ∧ ◡ PProd (F, G) Fn (C × D)))
7	6	an4s 799	. 2 ⊢ (((F Fn A ∧ ◡F Fn C) ∧ (G Fn B ∧ ◡G Fn D)) → ( PProd (F, G) Fn (A × B) ∧ ◡ PProd (F, G) Fn (C × D)))
8		dff1o4 5295	. . 3 ⊢ (F:A–1-1-onto→C ↔ (F Fn A ∧ ◡F Fn C))
9		dff1o4 5295	. . 3 ⊢ (G:B–1-1-onto→D ↔ (G Fn B ∧ ◡G Fn D))
10	8, 9	anbi12i 678	. 2 ⊢ ((F:A–1-1-onto→C ∧ G:B–1-1-onto→D) ↔ ((F Fn A ∧ ◡F Fn C) ∧ (G Fn B ∧ ◡G Fn D)))
11		dff1o4 5295	. 2 ⊢ ( PProd (F, G):(A × B)–1-1-onto→(C × D) ↔ ( PProd (F, G) Fn (A × B) ∧ ◡ PProd (F, G) Fn (C × D)))
12	7, 10, 11	3imtr4i 257	1 ⊢ ((F:A–1-1-onto→C ∧ G:B–1-1-onto→D) → PProd (F, G):(A × B)–1-1-onto→(C × D))

Syntax hints:   → wi 4   ∧ wa 358   × cxp 4771  ^◡ccnv 4772   Fn wfn 4777  –1-1-onto→wf1o 4781   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-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-pprod 5739

This theorem is referenced by:  xpen  6056