Theorem fparlem3 8139

Description: Lemma for fpar 8141. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fparlem3	⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (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 6276	. 2 ⊢ (◡(1st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))) = ∪ 𝑥 ∈ 𝐴 (◡(1st ↾ (V × V)) ∘ ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
2		inss1 4237	. . . . 5 ⊢ (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ dom 𝐹
3		fndm 6671	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 4026	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ 𝐴)
5		dfco2a 6266	. . . 4 ⊢ ((dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ 𝐴 → (𝐹 ∘ (1st ↾ (V × V))) = ∪ 𝑥 ∈ 𝐴 ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
6	4, 5	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∘ (1st ↾ (V × V))) = ∪ 𝑥 ∈ 𝐴 ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
7	6	coeq2d 5873	. 2 ⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = (◡(1st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8		inss1 4237	. . . . . . . . 9 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ dom ({(𝐹‘𝑥)} × ({𝑥} × V))
9		dmxpss 6191	. . . . . . . . 9 ⊢ dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ⊆ {(𝐹‘𝑥)}
10	8, 9	sstri 3993	. . . . . . . 8 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ {(𝐹‘𝑥)}
11		dfco2a 6266	. . . . . . . 8 ⊢ ((dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ {(𝐹‘𝑥)} → (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((◡(1st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})))
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((◡(1st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}))
13		fvex 6919	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
14		fparlem1 8137	. . . . . . . . . 10 ⊢ (◡(1st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15		sneq 4636	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → {𝑦} = {(𝐹‘𝑥)})
16	15	xpeq1d 5714	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ({𝑦} × V) = ({(𝐹‘𝑥)} × V))
17	14, 16	eqtrid 2789	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (◡(1st ↾ (V × V)) “ {𝑦}) = ({(𝐹‘𝑥)} × V))
18	15	imaeq2d 6078	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}))
19		df-ima 5698	. . . . . . . . . . 11 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)})
20		ssid 4006	. . . . . . . . . . . . . 14 ⊢ {(𝐹‘𝑥)} ⊆ {(𝐹‘𝑥)}
21		xpssres 6036	. . . . . . . . . . . . . 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 5948	. . . . . . . . . . . 12 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ran ({(𝐹‘𝑥)} × ({𝑥} × V))
24	13	snnz 4776	. . . . . . . . . . . . 13 ⊢ {(𝐹‘𝑥)} ≠ ∅
25		rnxp 6190	. . . . . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({𝑥} × V)
28	19, 27	eqtri 2765	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ({𝑥} × V)
29	18, 28	eqtrdi 2793	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
30	17, 29	xpeq12d 5716	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((◡(1st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V)))
31	13, 30	iunxsn 5091	. . . . . . 7 ⊢ ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((◡(1st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
32	12, 31	eqtri 2765	. . . . . 6 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
33	32	cnveqi 5885	. . . . 5 ⊢ ◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = ◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V))
34		cnvco 5896	. . . . 5 ⊢ ◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ ◡({(𝐹‘𝑥)} × ({𝑥} × V)))
35		cnvxp 6177	. . . . 5 ⊢ ◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
36	33, 34, 35	3eqtr3i 2773	. . . 4 ⊢ (◡(1st ↾ (V × V)) ∘ ◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
37		fparlem1 8137	. . . . . . . . 9 ⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
38	37	xpeq2i 5712	. . . . . . . 8 ⊢ ({(𝐹‘𝑥)} × (◡(1st ↾ (V × V)) “ {𝑥})) = ({(𝐹‘𝑥)} × ({𝑥} × V))
39		fnsnfv 6988	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
40	39	xpeq1d 5714	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × (◡(1st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})))
41	38, 40	eqtr3id 2791	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})))
42	41	cnveqd 5886	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ◡((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})))
43		cnvxp 6177	. . . . . 6 ⊢ ◡((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})) = ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
44	42, 43	eqtrdi 2793	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
45	44	coeq2d 5873	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (◡(1st ↾ (V × V)) ∘ ◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (◡(1st ↾ (V × V)) ∘ ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
46	36, 45	eqtr3id 2791	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = (◡(1st ↾ (V × V)) ∘ ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
47	46	iuneq2dv 5016	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = ∪ 𝑥 ∈ 𝐴 (◡(1st ↾ (V × V)) ∘ ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
48	1, 7, 47	3eqtr4a 2803	1 ⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626  ∪ ciun 4991   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688   ∘ ccom 5689   Fn wfn 6556  ‘cfv 6561  1^st c1st 8012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1st 8014  df-2nd 8015

This theorem is referenced by:  fpar  8141