Theorem fparlem3 8039

Description: Lemma for fpar 8041. (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 6199	. 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 4182	. . . . 5 ⊢ (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ dom 𝐹
3		fndm 6579	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 3972	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ 𝐴)
5		dfco2a 6188	. . . 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 5797	. 2 ⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = (◡(1st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8		inss1 4182	. . . . . . . . 9 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ dom ({(𝐹‘𝑥)} × ({𝑥} × V))
9		dmxpss 6113	. . . . . . . . 9 ⊢ dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ⊆ {(𝐹‘𝑥)}
10	8, 9	sstri 3939	. . . . . . . 8 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ {(𝐹‘𝑥)}
11		dfco2a 6188	. . . . . . . 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 6830	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
14		fparlem1 8037	. . . . . . . . . 10 ⊢ (◡(1st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15		sneq 4581	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → {𝑦} = {(𝐹‘𝑥)})
16	15	xpeq1d 5640	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ({𝑦} × V) = ({(𝐹‘𝑥)} × V))
17	14, 16	eqtrid 2778	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (◡(1st ↾ (V × V)) “ {𝑦}) = ({(𝐹‘𝑥)} × V))
18	15	imaeq2d 6004	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}))
19		df-ima 5624	. . . . . . . . . . 11 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)})
20		ssid 3952	. . . . . . . . . . . . . 14 ⊢ {(𝐹‘𝑥)} ⊆ {(𝐹‘𝑥)}
21		xpssres 5962	. . . . . . . . . . . . . 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 5872	. . . . . . . . . . . 12 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ran ({(𝐹‘𝑥)} × ({𝑥} × V))
24	13	snnz 4724	. . . . . . . . . . . . 13 ⊢ {(𝐹‘𝑥)} ≠ ∅
25		rnxp 6112	. . . . . . . . . . . . 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 2754	. . . . . . . . . . 11 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({𝑥} × V)
28	19, 27	eqtri 2754	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ({𝑥} × V)
29	18, 28	eqtrdi 2782	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
30	17, 29	xpeq12d 5642	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((◡(1st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V)))
31	13, 30	iunxsn 5034	. . . . . . 7 ⊢ ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((◡(1st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
32	12, 31	eqtri 2754	. . . . . 6 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
33	32	cnveqi 5809	. . . . 5 ⊢ ◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = ◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V))
34		cnvco 5820	. . . . 5 ⊢ ◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ ◡({(𝐹‘𝑥)} × ({𝑥} × V)))
35		cnvxp 6099	. . . . 5 ⊢ ◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
36	33, 34, 35	3eqtr3i 2762	. . . 4 ⊢ (◡(1st ↾ (V × V)) ∘ ◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
37		fparlem1 8037	. . . . . . . . 9 ⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
38	37	xpeq2i 5638	. . . . . . . 8 ⊢ ({(𝐹‘𝑥)} × (◡(1st ↾ (V × V)) “ {𝑥})) = ({(𝐹‘𝑥)} × ({𝑥} × V))
39		fnsnfv 6896	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
40	39	xpeq1d 5640	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × (◡(1st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})))
41	38, 40	eqtr3id 2780	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})))
42	41	cnveqd 5810	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ◡((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})))
43		cnvxp 6099	. . . . . 6 ⊢ ◡((𝐹 “ {𝑥}) × (◡(1st ↾ (V × V)) “ {𝑥})) = ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
44	42, 43	eqtrdi 2782	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
45	44	coeq2d 5797	. . . 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 2780	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = (◡(1st ↾ (V × V)) ∘ ((◡(1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
47	46	iuneq2dv 4961	. 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 2792	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 1541   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  ∅c0 4278  {csn 4571  ∪ ciun 4936   × cxp 5609  ^◡ccnv 5610  dom cdm 5611  ran crn 5612   ↾ cres 5613   “ cima 5614   ∘ ccom 5615   Fn wfn 6471  ‘cfv 6476  1^st c1st 7914

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-1st 7916  df-2nd 7917

This theorem is referenced by:  fpar  8041