Theorem fnpprod 5844

Description: Functionhood law for parallel product. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
fnpprod	⊢ ((F Fn A ∧ G Fn B) → PProd (F, G) Fn (A × B))

Proof of Theorem fnpprod

Dummy variables a b c d e f g h x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ee4anv 1915	. . . . . . . . 9 ⊢ (∃a∃b∃e∃f(∃c∃d(x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ ∃g∃h(x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))) ↔ (∃a∃b∃c∃d(x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ ∃e∃f∃g∃h(x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))))
2		ee4anv 1915	. . . . . . . . . . 11 ⊢ (∃c∃d∃g∃h((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ (x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))) ↔ (∃c∃d(x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ ∃g∃h(x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))))
3	2	2exbii 1583	. . . . . . . . . 10 ⊢ (∃e∃f∃c∃d∃g∃h((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ (x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))) ↔ ∃e∃f(∃c∃d(x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ ∃g∃h(x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))))
4	3	2exbii 1583	. . . . . . . . 9 ⊢ (∃a∃b∃e∃f∃c∃d∃g∃h((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ (x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))) ↔ ∃a∃b∃e∃f(∃c∃d(x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ ∃g∃h(x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))))
5		brpprod 5840	. . . . . . . . . 10 ⊢ (x PProd (F, G)y ↔ ∃a∃b∃c∃d(x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)))
6		brpprod 5840	. . . . . . . . . 10 ⊢ (x PProd (F, G)z ↔ ∃e∃f∃g∃h(x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)))
7	5, 6	anbi12i 678	. . . . . . . . 9 ⊢ ((x PProd (F, G)y ∧ x PProd (F, G)z) ↔ (∃a∃b∃c∃d(x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ ∃e∃f∃g∃h(x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))))
8	1, 4, 7	3bitr4ri 269	. . . . . . . 8 ⊢ ((x PProd (F, G)y ∧ x PProd (F, G)z) ↔ ∃a∃b∃e∃f∃c∃d∃g∃h((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ (x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))))
9		an42 798	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((aFc ∧ bGd) ∧ (aFg ∧ bGh)) ↔ ((aFc ∧ aFg) ∧ (bGh ∧ bGd)))
10		fununiq 5518	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun F ∧ aFc ∧ aFg) → c = g)
11	10	3expib 1154	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Fun F → ((aFc ∧ aFg) → c = g))
12		fununiq 5518	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Fun G ∧ bGh ∧ bGd) → h = d)
13	12	eqcomd 2358	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun G ∧ bGh ∧ bGd) → d = h)
14	13	3expib 1154	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Fun G → ((bGh ∧ bGd) → d = h))
15	11, 14	im2anan9 808	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun F ∧ Fun G) → (((aFc ∧ aFg) ∧ (bGh ∧ bGd)) → (c = g ∧ d = h)))
16	9, 15	syl5bi 208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun F ∧ Fun G) → (((aFc ∧ bGd) ∧ (aFg ∧ bGh)) → (c = g ∧ d = h)))
17	16	exp3acom23 1372	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun F ∧ Fun G) → ((aFg ∧ bGh) → ((aFc ∧ bGd) → (c = g ∧ d = h))))
18		breq1 4643	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (e = a → (eFg ↔ aFg))
19		breq1 4643	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (f = b → (fGh ↔ bGh))
20	18, 19	bi2anan9 843	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((e = a ∧ f = b) → ((eFg ∧ fGh) ↔ (aFg ∧ bGh)))
21	20	adantr 451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((e = a ∧ f = b) ∧ z = ⟨g, h⟩) → ((eFg ∧ fGh) ↔ (aFg ∧ bGh)))
22		eqeq2 2362	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (z = ⟨g, h⟩ → (⟨c, d⟩ = z ↔ ⟨c, d⟩ = ⟨g, h⟩))
23		opth 4603	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨c, d⟩ = ⟨g, h⟩ ↔ (c = g ∧ d = h))
24	22, 23	syl6bb 252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (z = ⟨g, h⟩ → (⟨c, d⟩ = z ↔ (c = g ∧ d = h)))
25	24	imbi2d 307	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (z = ⟨g, h⟩ → (((aFc ∧ bGd) → ⟨c, d⟩ = z) ↔ ((aFc ∧ bGd) → (c = g ∧ d = h))))
26	25	adantl 452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((e = a ∧ f = b) ∧ z = ⟨g, h⟩) → (((aFc ∧ bGd) → ⟨c, d⟩ = z) ↔ ((aFc ∧ bGd) → (c = g ∧ d = h))))
27	21, 26	imbi12d 311	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((e = a ∧ f = b) ∧ z = ⟨g, h⟩) → (((eFg ∧ fGh) → ((aFc ∧ bGd) → ⟨c, d⟩ = z)) ↔ ((aFg ∧ bGh) → ((aFc ∧ bGd) → (c = g ∧ d = h)))))
28	17, 27	syl5ibrcom 213	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun F ∧ Fun G) → (((e = a ∧ f = b) ∧ z = ⟨g, h⟩) → ((eFg ∧ fGh) → ((aFc ∧ bGd) → ⟨c, d⟩ = z))))
29	28	exp3a 425	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun F ∧ Fun G) → ((e = a ∧ f = b) → (z = ⟨g, h⟩ → ((eFg ∧ fGh) → ((aFc ∧ bGd) → ⟨c, d⟩ = z)))))
30	29	3impd 1165	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun F ∧ Fun G) → (((e = a ∧ f = b) ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) → ((aFc ∧ bGd) → ⟨c, d⟩ = z)))
31	30	com23 72	. . . . . . . . . . . . . . . 16 ⊢ ((Fun F ∧ Fun G) → ((aFc ∧ bGd) → (((e = a ∧ f = b) ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) → ⟨c, d⟩ = z)))
32		eqeq1 2359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (x = ⟨a, b⟩ → (x = ⟨e, f⟩ ↔ ⟨a, b⟩ = ⟨e, f⟩))
33		eqcom 2355	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨a, b⟩ = ⟨e, f⟩ ↔ ⟨e, f⟩ = ⟨a, b⟩)
34		opth 4603	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨e, f⟩ = ⟨a, b⟩ ↔ (e = a ∧ f = b))
35	33, 34	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨a, b⟩ = ⟨e, f⟩ ↔ (e = a ∧ f = b))
36	32, 35	syl6bb 252	. . . . . . . . . . . . . . . . . . . 20 ⊢ (x = ⟨a, b⟩ → (x = ⟨e, f⟩ ↔ (e = a ∧ f = b)))
37	36	3anbi1d 1256	. . . . . . . . . . . . . . . . . . 19 ⊢ (x = ⟨a, b⟩ → ((x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) ↔ ((e = a ∧ f = b) ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))))
38	37	adantr 451	. . . . . . . . . . . . . . . . . 18 ⊢ ((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩) → ((x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) ↔ ((e = a ∧ f = b) ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))))
39		eqeq1 2359	. . . . . . . . . . . . . . . . . . 19 ⊢ (y = ⟨c, d⟩ → (y = z ↔ ⟨c, d⟩ = z))
40	39	adantl 452	. . . . . . . . . . . . . . . . . 18 ⊢ ((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩) → (y = z ↔ ⟨c, d⟩ = z))
41	38, 40	imbi12d 311	. . . . . . . . . . . . . . . . 17 ⊢ ((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩) → (((x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) → y = z) ↔ (((e = a ∧ f = b) ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) → ⟨c, d⟩ = z)))
42	41	imbi2d 307	. . . . . . . . . . . . . . . 16 ⊢ ((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩) → (((aFc ∧ bGd) → ((x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) → y = z)) ↔ ((aFc ∧ bGd) → (((e = a ∧ f = b) ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) → ⟨c, d⟩ = z))))
43	31, 42	syl5ibrcom 213	. . . . . . . . . . . . . . 15 ⊢ ((Fun F ∧ Fun G) → ((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩) → ((aFc ∧ bGd) → ((x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) → y = z))))
44	43	exp3a 425	. . . . . . . . . . . . . 14 ⊢ ((Fun F ∧ Fun G) → (x = ⟨a, b⟩ → (y = ⟨c, d⟩ → ((aFc ∧ bGd) → ((x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) → y = z)))))
45	44	3impd 1165	. . . . . . . . . . . . 13 ⊢ ((Fun F ∧ Fun G) → ((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) → ((x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh)) → y = z)))
46	45	imp3a 420	. . . . . . . . . . . 12 ⊢ ((Fun F ∧ Fun G) → (((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ (x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))) → y = z))
47	46	exlimdvv 1637	. . . . . . . . . . 11 ⊢ ((Fun F ∧ Fun G) → (∃g∃h((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ (x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))) → y = z))
48	47	exlimdvv 1637	. . . . . . . . . 10 ⊢ ((Fun F ∧ Fun G) → (∃c∃d∃g∃h((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ (x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))) → y = z))
49	48	exlimdvv 1637	. . . . . . . . 9 ⊢ ((Fun F ∧ Fun G) → (∃e∃f∃c∃d∃g∃h((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ (x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))) → y = z))
50	49	exlimdvv 1637	. . . . . . . 8 ⊢ ((Fun F ∧ Fun G) → (∃a∃b∃e∃f∃c∃d∃g∃h((x = ⟨a, b⟩ ∧ y = ⟨c, d⟩ ∧ (aFc ∧ bGd)) ∧ (x = ⟨e, f⟩ ∧ z = ⟨g, h⟩ ∧ (eFg ∧ fGh))) → y = z))
51	8, 50	syl5bi 208	. . . . . . 7 ⊢ ((Fun F ∧ Fun G) → ((x PProd (F, G)y ∧ x PProd (F, G)z) → y = z))
52	51	alrimiv 1631	. . . . . 6 ⊢ ((Fun F ∧ Fun G) → ∀z((x PProd (F, G)y ∧ x PProd (F, G)z) → y = z))
53	52	alrimivv 1632	. . . . 5 ⊢ ((Fun F ∧ Fun G) → ∀x∀y∀z((x PProd (F, G)y ∧ x PProd (F, G)z) → y = z))
54		dffun2 5120	. . . . 5 ⊢ (Fun PProd (F, G) ↔ ∀x∀y∀z((x PProd (F, G)y ∧ x PProd (F, G)z) → y = z))
55	53, 54	sylibr 203	. . . 4 ⊢ ((Fun F ∧ Fun G) → Fun PProd (F, G))
56		dmpprod 5841	. . . . 5 ⊢ dom PProd (F, G) = (dom F × dom G)
57		xpeq12 4804	. . . . 5 ⊢ ((dom F = A ∧ dom G = B) → (dom F × dom G) = (A × B))
58	56, 57	syl5eq 2397	. . . 4 ⊢ ((dom F = A ∧ dom G = B) → dom PProd (F, G) = (A × B))
59	55, 58	anim12i 549	. . 3 ⊢ (((Fun F ∧ Fun G) ∧ (dom F = A ∧ dom G = B)) → (Fun PProd (F, G) ∧ dom PProd (F, G) = (A × B)))
60	59	an4s 799	. 2 ⊢ (((Fun F ∧ dom F = A) ∧ (Fun G ∧ dom G = B)) → (Fun PProd (F, G) ∧ dom PProd (F, G) = (A × B)))
61		df-fn 4791	. . 3 ⊢ (F Fn A ↔ (Fun F ∧ dom F = A))
62		df-fn 4791	. . 3 ⊢ (G Fn B ↔ (Fun G ∧ dom G = B))
63	61, 62	anbi12i 678	. 2 ⊢ ((F Fn A ∧ G Fn B) ↔ ((Fun F ∧ dom F = A) ∧ (Fun G ∧ dom G = B)))
64		df-fn 4791	. 2 ⊢ ( PProd (F, G) Fn (A × B) ↔ (Fun PProd (F, G) ∧ dom PProd (F, G) = (A × B)))
65	60, 63, 64	3imtr4i 257	1 ⊢ ((F Fn A ∧ G Fn B) → PProd (F, G) Fn (A × B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642  ⟨cop 4562   class class class wbr 4640   × cxp 4771  dom cdm 4773  Fun wfun 4776   Fn wfn 4777   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-2nd 4798  df-txp 5737  df-pprod 5739

This theorem is referenced by:  f1opprod  5845