Theorem fmucndlem 22095

Description: Lemma for fmucnd 22096. (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 5127	. . 3 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ran ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴))
2		simpr 477	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
3		resmpt2 6758	. . . . 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 701	. . . 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 5353	. . 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	syl5eq 2668	. 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 3203	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	op1std 7178	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑝) = 𝑥)
10	9	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑝)) = (𝐹‘𝑥))
11	7, 8	op2ndd 7179	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑝) = 𝑦)
12	11	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑝)) = (𝐹‘𝑦))
13	10, 12	opeq12d 4410	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
14	13	mpt2mpt 6752	. . . . . . . . 9 ⊢ (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
15	14	eqcomi 2631	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
16	15	rneqi 5352	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = ran (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
17		fvexd 6203	. . . . . . 7 ⊢ ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(1st ‘𝑝)) ∈ V)
18		fvexd 6203	. . . . . . 7 ⊢ ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(2nd ‘𝑝)) ∈ V)
19	16, 17, 18	fliftrel 6558	. . . . . 6 ⊢ (⊤ → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⊆ (V × V))
20	19	trud 1493	. . . . 5 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⊆ (V × V)
21	20	sseli 3599	. . . 4 ⊢ (𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) → 𝑝 ∈ (V × V))
22	21	adantl 482	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)) → 𝑝 ∈ (V × V))
23		xpss 5226	. . . . 5 ⊢ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ⊆ (V × V)
24	23	sseli 3599	. . . 4 ⊢ (𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) → 𝑝 ∈ (V × V))
25	24	adantl 482	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))) → 𝑝 ∈ (V × V))
26		fvelimab 6253	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝)))
27		fvelimab 6253	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((2nd ‘𝑝) ∈ (𝐹 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝)))
28	26, 27	anbi12d 747	. . . . . . 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))))
29		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
30		opex 4932	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ V
31	29, 30	elrnmpt2 6773	. . . . . . . 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)⟩)
32		eqcom 2629	. . . . . . . . . 10 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
33		fvex 6201	. . . . . . . . . . 11 ⊢ (1st ‘𝑝) ∈ V
34		fvex 6201	. . . . . . . . . . 11 ⊢ (2nd ‘𝑝) ∈ V
35	33, 34	opth2 4949	. . . . . . . . . 10 ⊢ (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ↔ ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
36	32, 35	bitri 264	. . . . . . . . 9 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
37	36	2rexbii 3042	. . . . . . . 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)))
38		reeanv 3107	. . . . . . . 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)))
39	31, 37, 38	3bitri 286	. . . . . . 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)))
40	28, 39	syl6rbbr 279	. . . . . 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 5146	. . . . . 6 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ↔ ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ∧ (2nd ‘𝑝) ∈ (𝐹 “ 𝐴)))
42	40, 41	syl6bbr 278	. . . . 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 481	. . . 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 7205	. . . . . 6 ⊢ (𝑝 ∈ (V × V) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
45	44	adantl 482	. . . . 5 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
46	45	eleq1d 2686	. . . 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 2686	. . . 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 300	. . 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 2621	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)))
50	6, 49	eqtrd 2656	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 196   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ⟨cop 4183   ↦ cmpt 4729   × cxp 5112  ran crn 5115   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ‘cfv 5888   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  fmucnd  22096