Theorem fmucndlem 24017

Description: Lemma for fmucnd 24018. (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 5689	. . 3 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ran ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴))
2		simpr 484	. . . . 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 587	. . . 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 5937	. . 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 2783	. 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 3477	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 3477	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	op1std 7989	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑝) = 𝑥)
10	9	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑝)) = (𝐹‘𝑥))
11	7, 8	op2ndd 7990	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑝) = 𝑦)
12	11	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑝)) = (𝐹‘𝑦))
13	10, 12	opeq12d 4881	. . . . . . . . . 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 2740	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
16	15	rneqi 5936	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = ran (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
17		fvexd 6906	. . . . . . 7 ⊢ ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(1st ‘𝑝)) ∈ V)
18		fvexd 6906	. . . . . . 7 ⊢ ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(2nd ‘𝑝)) ∈ V)
19	16, 17, 18	fliftrel 7308	. . . . . 6 ⊢ (⊤ → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⊆ (V × V))
20	19	mptru 1547	. . . . 5 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⊆ (V × V)
21	20	sseli 3978	. . . 4 ⊢ (𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) → 𝑝 ∈ (V × V))
22	21	adantl 481	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)) → 𝑝 ∈ (V × V))
23		xpss 5692	. . . . 5 ⊢ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ⊆ (V × V)
24	23	sseli 3978	. . . 4 ⊢ (𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) → 𝑝 ∈ (V × V))
25	24	adantl 481	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))) → 𝑝 ∈ (V × V))
26		eqid 2731	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
27		opex 5464	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ V
28	26, 27	elrnmpo 7548	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
30		fvex 6904	. . . . . . . . . . 11 ⊢ (1st ‘𝑝) ∈ V
31		fvex 6904	. . . . . . . . . . 11 ⊢ (2nd ‘𝑝) ∈ V
32	30, 31	opth2 5480	. . . . . . . . . 10 ⊢ (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ↔ ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
33	29, 32	bitri 275	. . . . . . . . 9 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
34	33	2rexbii 3128	. . . . . . . 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 3225	. . . . . . . 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 297	. . . . . . 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 6964	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝)))
38		fvelimab 6964	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((2nd ‘𝑝) ∈ (𝐹 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝)))
39	37, 38	anbi12d 630	. . . . . . 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 290	. . . . . 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 5712	. . . . . 6 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ↔ ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ∧ (2nd ‘𝑝) ∈ (𝐹 “ 𝐴)))
42	40, 41	bitr4di 289	. . . . 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 480	. . . 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 8018	. . . . . 6 ⊢ (𝑝 ∈ (V × V) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
45	44	adantl 481	. . . . 5 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
46	45	eleq1d 2817	. . . 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 2817	. . . 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 311	. . 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 2730	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)))
50	6, 49	eqtrd 2771	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 205   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2105  ∃wrex 3069  Vcvv 3473   ⊆ wss 3948  ⟨cop 4634   ↦ cmpt 5231   × cxp 5674  ran crn 5677   ↾ cres 5678   “ cima 5679   Fn wfn 6538  ‘cfv 6543   ∈ cmpo 7414  1^st c1st 7977  2^nd c2nd 7978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980

This theorem is referenced by:  fmucnd  24018