Theorem fmucndlem 24415

Description: Lemma for fmucnd 24416. (Contributed by Thierry Arnoux, 19-Nov-2017.)

Assertion

Ref	Expression
fmucndlem	⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦

Proof of Theorem fmucndlem

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 5675	. . 3 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ran ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴))
2		simpr 489	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
3		resmpo 7531	. . . . 5 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩))
4	2, 3	sylancom 599	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩))
5	4	rneqd 5929	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ran ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴)) = ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩))
6	1, 5	eqtrid 2816	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩))
7		vex 3467	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 3467	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	op1std 7995	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑝) = 𝑥)
10	9	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑝)) = (𝐹‘𝑥))
11	7, 8	op2ndd 7996	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑝) = 𝑦)
12	11	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑝)) = (𝐹‘𝑦))
13	10, 12	opeq12d 4850	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
14	13	mpompt 7525	. . . . . . . . 9 ⊢ (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
15	14	eqcomi 2778	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
16	15	rneqi 5928	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = ran (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
17		fvexd 6897	. . . . . . 7 ⊢ ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(1st ‘𝑝)) ∈ V)
18		fvexd 6897	. . . . . . 7 ⊢ ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(2nd ‘𝑝)) ∈ V)
19	16, 17, 18	fliftrel 7307	. . . . . 6 ⊢ (⊤ → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⊆ (V × V))
20	19	mptru 1574	. . . . 5 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⊆ (V × V)
21	20	sseli 3941	. . . 4 ⊢ (𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) → 𝑝 ∈ (V × V))
22	21	adantl 486	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)) → 𝑝 ∈ (V × V))
23		xpss 5678	. . . . 5 ⊢ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ⊆ (V × V)
24	23	sseli 3941	. . . 4 ⊢ (𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) → 𝑝 ∈ (V × V))
25	24	adantl 486	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))) → 𝑝 ∈ (V × V))
26		eqid 2769	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
27		opex 5446	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ V
28	26, 27	elrnmpo 7547	. . . . . . . 8 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
29		eqcom 2776	. . . . . . . . . 10 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
30		fvex 6895	. . . . . . . . . . 11 ⊢ (1st ‘𝑝) ∈ V
31		fvex 6895	. . . . . . . . . . 11 ⊢ (2nd ‘𝑝) ∈ V
32	30, 31	opth2 5463	. . . . . . . . . 10 ⊢ (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ↔ ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
33	29, 32	bitri 278	. . . . . . . . 9 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
34	33	2rexbii 3147	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
35		reeanv 3243	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)) ↔ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝) ∧ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝)))
36	28, 34, 35	3bitri 300	. . . . . . 7 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝) ∧ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝)))
37		fvelimab 6954	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝)))
38		fvelimab 6954	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((2nd ‘𝑝) ∈ (𝐹 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝)))
39	37, 38	anbi12d 643	. . . . . . 7 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → (((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ∧ (2nd ‘𝑝) ∈ (𝐹 “ 𝐴)) ↔ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝) ∧ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝))))
40	36, 39	bitr4id 293	. . . . . 6 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ∧ (2nd ‘𝑝) ∈ (𝐹 “ 𝐴))))
41		opelxp 5698	. . . . . 6 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ↔ ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ∧ (2nd ‘𝑝) ∈ (𝐹 “ 𝐴)))
42	40, 41	bitr4di 292	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))))
43	42	adantr 485	. . . 4 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))))
44		1st2nd2 8024	. . . . . 6 ⊢ (𝑝 ∈ (V × V) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
45	44	adantl 486	. . . . 5 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
46	45	eleq1d 2854	. . . 4 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → (𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)))
47	45	eleq1d 2854	. . . 4 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → (𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ↔ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))))
48	43, 46, 47	3bitr4d 314	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → (𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ 𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))))
49	22, 25, 48	eqrdav 2768	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)))
50	6, 49	eqtrd 2804	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ⊤wtru 1568   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ⟨cop 4600   ↦ cmpt 5196   × cxp 5660  ran crn 5663   ↾ cres 5664   “ cima 5665   Fn wfn 6532  ‘cfv 6537   ∈ cmpo 7413  1^st c1st 7983  2^nd c2nd 7984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986

This theorem is referenced by:  fmucnd  24416