fparlem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fparlem3		Structured version Visualization version GIF version

Theorem fparlem3 8105

Description: Lemma for fpar 8107. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Assertion**
Ref	Expression
fparlem3	⊢ (𝐹 Fn 𝐴 → (^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐹

Proof of Theorem fparlem3 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coiun 6256	. 2 ⊢ (^◡(1^st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))) = ∪ 𝑥 ∈ 𝐴 (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
2		inss1 4197	. . . . 5 ⊢ (dom 𝐹 ∩ ran (1^st ↾ (V × V))) ⊆ dom 𝐹
3		fndm 6636	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 3987	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1^st ↾ (V × V))) ⊆ 𝐴)
5		dfco2a 6245	. . . 4 ⊢ ((dom 𝐹 ∩ ran (1^st ↾ (V × V))) ⊆ 𝐴 → (𝐹 ∘ (1^st ↾ (V × V))) = ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
6	4, 5	syl 18	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∘ (1^st ↾ (V × V))) = ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
7	6	coeq2d 5846	. 2 ⊢ (𝐹 Fn 𝐴 → (^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) = (^◡(1^st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8		inss1 4197	. . . . . . . . 9 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1^st ↾ (V × V))) ⊆ dom ({(𝐹‘𝑥)} × ({𝑥} × V))
9		dmxpss 6167	. . . . . . . . 9 ⊢ dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ⊆ {(𝐹‘𝑥)}
10	8, 9	sstri 3954	. . . . . . . 8 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1^st ↾ (V × V))) ⊆ {(𝐹‘𝑥)}
11		dfco2a 6245	. . . . . . . 8 ⊢ ((dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1^st ↾ (V × V))) ⊆ {(𝐹‘𝑥)} → (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})))
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}))
13		fvex 6892	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
14		fparlem1 8103	. . . . . . . . . 10 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15		sneq 4601	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → {𝑦} = {(𝐹‘𝑥)})
16	15	xpeq1d 5688	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ({𝑦} × V) = ({(𝐹‘𝑥)} × V))
17	14, 16	eqtrid 2816	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (^◡(1^st ↾ (V × V)) “ {𝑦}) = ({(𝐹‘𝑥)} × V))
18	15	imaeq2d 6060	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}))
19		df-ima 5672	. . . . . . . . . . 11 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)})
20		ssid 3967	. . . . . . . . . . . . . 14 ⊢ {(𝐹‘𝑥)} ⊆ {(𝐹‘𝑥)}
21		xpssres 6015	. . . . . . . . . . . . . 14 ⊢ ({(𝐹‘𝑥)} ⊆ {(𝐹‘𝑥)} → (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({(𝐹‘𝑥)} × ({𝑥} × V)))
22	20, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({(𝐹‘𝑥)} × ({𝑥} × V))
23	22	rneqi 5925	. . . . . . . . . . . 12 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ran ({(𝐹‘𝑥)} × ({𝑥} × V))
24	13	snnz 4744	. . . . . . . . . . . . 13 ⊢ {(𝐹‘𝑥)} ≠ ∅
25		rnxp 6166	. . . . . . . . . . . . 13 ⊢ ({(𝐹‘𝑥)} ≠ ∅ → ran ({(𝐹‘𝑥)} × ({𝑥} × V)) = ({𝑥} × V))
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran ({(𝐹‘𝑥)} × ({𝑥} × V)) = ({𝑥} × V)
27	23, 26	eqtri 2792	. . . . . . . . . . 11 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({𝑥} × V)
28	19, 27	eqtri 2792	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ({𝑥} × V)
29	18, 28	eqtrdi 2820	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
30	17, 29	xpeq12d 5690	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V)))
31	13, 30	iunxsn 5058	. . . . . . 7 ⊢ ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
32	12, 31	eqtri 2792	. . . . . 6 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
33	32	cnveqi 5858	. . . . 5 ⊢ ^◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = ^◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V))
34		cnvco 5873	. . . . 5 ⊢ ^◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = (^◡(1^st ↾ (V × V)) ∘ ^◡({(𝐹‘𝑥)} × ({𝑥} × V)))
35		cnvxp 6152	. . . . 5 ⊢ ^◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
36	33, 34, 35	3eqtr3i 2800	. . . 4 ⊢ (^◡(1^st ↾ (V × V)) ∘ ^◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
37		fparlem1 8103	. . . . . . . . 9 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
38	37	xpeq2i 5686	. . . . . . . 8 ⊢ ({(𝐹‘𝑥)} × (^◡(1^st ↾ (V × V)) “ {𝑥})) = ({(𝐹‘𝑥)} × ({𝑥} × V))
39		fnsnfv 6958	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
40	39	xpeq1d 5688	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × (^◡(1^st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})))
41	38, 40	eqtr3id 2818	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})))
42	41	cnveqd 5859	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ^◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ^◡((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})))
43		cnvxp 6152	. . . . . 6 ⊢ ^◡((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})) = ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
44	42, 43	eqtrdi 2820	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ^◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
45	44	coeq2d 5846	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (^◡(1^st ↾ (V × V)) ∘ ^◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
46	36, 45	eqtr3id 2818	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
47	46	iuneq2dv 4982	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = ∪ 𝑥 ∈ 𝐴 (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
48	1, 7, 47	3eqtr4a 2830	1 ⊢ (𝐹 Fn 𝐴 → (^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 Vcvv 3463 ∩ cin 3912 ⊆ wss 3913 ∅c0 4294 {csn 4591 ∪ ciun 4957 × cxp 5657 ^◡ccnv 5658 dom cdm 5659 ran crn 5660 ↾ cres 5661 “ cima 5662 ∘ ccom 5663 Fn wfn 6529 ‘cfv 6534 1^st c1st 7980

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 5258 ax-nul 5268 ax-pr 5402 ax-un 7730

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 4490 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-1st 7982 df-2nd 7983

This theorem is referenced by: fpar 8107

W3C validator