Theorem fparlem3 8102

Description: Lemma for fpar 8104. (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 6254	. 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 4227	. . . . 5 ⊢ (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ dom 𝐹
3		fndm 6651	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 4033	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ 𝐴)
5		dfco2a 6244	. . . 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 5861	. 2 ⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = (◡(1st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8		inss1 4227	. . . . . . . . 9 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ dom ({(𝐹‘𝑥)} × ({𝑥} × V))
9		dmxpss 6169	. . . . . . . . 9 ⊢ dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ⊆ {(𝐹‘𝑥)}
10	8, 9	sstri 3990	. . . . . . . 8 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ {(𝐹‘𝑥)}
11		dfco2a 6244	. . . . . . . 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 6903	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
14		fparlem1 8100	. . . . . . . . . 10 ⊢ (◡(1st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15		sneq 4637	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → {𝑦} = {(𝐹‘𝑥)})
16	15	xpeq1d 5704	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ({𝑦} × V) = ({(𝐹‘𝑥)} × V))
17	14, 16	eqtrid 2782	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (◡(1st ↾ (V × V)) “ {𝑦}) = ({(𝐹‘𝑥)} × V))
18	15	imaeq2d 6058	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}))
19		df-ima 5688	. . . . . . . . . . 11 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)})
20		ssid 4003	. . . . . . . . . . . . . 14 ⊢ {(𝐹‘𝑥)} ⊆ {(𝐹‘𝑥)}
21		xpssres 6017	. . . . . . . . . . . . . 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 5935	. . . . . . . . . . . 12 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ran ({(𝐹‘𝑥)} × ({𝑥} × V))
24	13	snnz 4779	. . . . . . . . . . . . 13 ⊢ {(𝐹‘𝑥)} ≠ ∅
25		rnxp 6168	. . . . . . . . . . . . 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 2758	. . . . . . . . . . 11 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({𝑥} × V)
28	19, 27	eqtri 2758	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ({𝑥} × V)
29	18, 28	eqtrdi 2786	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
30	17, 29	xpeq12d 5706	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((◡(1st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V)))
31	13, 30	iunxsn 5093	. . . . . . 7 ⊢ ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((◡(1st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
32	12, 31	eqtri 2758	. . . . . 6 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
33	32	cnveqi 5873	. . . . 5 ⊢ ◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = ◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V))
34		cnvco 5884	. . . . 5 ⊢ ◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ ◡({(𝐹‘𝑥)} × ({𝑥} × V)))
35		cnvxp 6155	. . . . 5 ⊢ ◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
36	33, 34, 35	3eqtr3i 2766	. . . 4 ⊢ (◡(1st ↾ (V × V)) ∘ ◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
37		fparlem1 8100	. . . . . . . . 9 ⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
38	37	xpeq2i 5702	. . . . . . . 8 ⊢ ({(𝐹‘𝑥)} × (◡(1st ↾ (V × V)) “ {𝑥})) = ({(𝐹‘𝑥)} × ({𝑥} × V))
39		fnsnfv 6969	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
40	39	xpeq1d 5704	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × (◡(1st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})))
41	38, 40	eqtr3id 2784	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})))
42	41	cnveqd 5874	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ◡((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})))
43		cnvxp 6155	. . . . . 6 ⊢ ◡((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})) = ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
44	42, 43	eqtrdi 2786	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
45	44	coeq2d 5861	. . . 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 2784	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = (◡(1st ↾ (V × V)) ∘ ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
47	46	iuneq2dv 5020	. 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 2796	1 ⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  Vcvv 3472   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  {csn 4627  ∪ ciun 4996   × cxp 5673  ^◡ccnv 5674  dom cdm 5675  ran crn 5676   ↾ cres 5677   “ cima 5678   ∘ ccom 5679   Fn wfn 6537  ‘cfv 6542  1^st c1st 7975

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-1st 7977  df-2nd 7978

This theorem is referenced by:  fpar  8104