Theorem fparlem4 8097

Description: Lemma for fpar 8098. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fparlem4	⊢ (𝐺 Fn 𝐵 → (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)})))

Distinct variable groups:   𝑦,𝐵   𝑦,𝐺

Proof of Theorem fparlem4

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coiun 6232	. 2 ⊢ (◡(2nd ↾ (V × V)) ∘ ∪ 𝑦 ∈ 𝐵 ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))) = ∪ 𝑦 ∈ 𝐵 (◡(2nd ↾ (V × V)) ∘ ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
2		inss1 4203	. . . . 5 ⊢ (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ dom 𝐺
3		fndm 6624	. . . . 5 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
4	2, 3	sseqtrid 3992	. . . 4 ⊢ (𝐺 Fn 𝐵 → (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ 𝐵)
5		dfco2a 6222	. . . 4 ⊢ ((dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ 𝐵 → (𝐺 ∘ (2nd ↾ (V × V))) = ∪ 𝑦 ∈ 𝐵 ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
6	4, 5	syl 17	. . 3 ⊢ (𝐺 Fn 𝐵 → (𝐺 ∘ (2nd ↾ (V × V))) = ∪ 𝑦 ∈ 𝐵 ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
7	6	coeq2d 5829	. 2 ⊢ (𝐺 Fn 𝐵 → (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = (◡(2nd ↾ (V × V)) ∘ ∪ 𝑦 ∈ 𝐵 ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
8		inss1 4203	. . . . . . . . 9 ⊢ (dom ({(𝐺‘𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ dom ({(𝐺‘𝑦)} × (V × {𝑦}))
9		dmxpss 6147	. . . . . . . . 9 ⊢ dom ({(𝐺‘𝑦)} × (V × {𝑦})) ⊆ {(𝐺‘𝑦)}
10	8, 9	sstri 3959	. . . . . . . 8 ⊢ (dom ({(𝐺‘𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ {(𝐺‘𝑦)}
11		dfco2a 6222	. . . . . . . 8 ⊢ ((dom ({(𝐺‘𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ {(𝐺‘𝑦)} → (({(𝐺‘𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ∪ 𝑥 ∈ {(𝐺‘𝑦)} ((◡(2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺‘𝑦)} × (V × {𝑦})) “ {𝑥})))
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ (({(𝐺‘𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ∪ 𝑥 ∈ {(𝐺‘𝑦)} ((◡(2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺‘𝑦)} × (V × {𝑦})) “ {𝑥}))
13		fvex 6874	. . . . . . . 8 ⊢ (𝐺‘𝑦) ∈ V
14		fparlem2 8095	. . . . . . . . . 10 ⊢ (◡(2nd ↾ (V × V)) “ {𝑥}) = (V × {𝑥})
15		sneq 4602	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑦) → {𝑥} = {(𝐺‘𝑦)})
16	15	xpeq2d 5671	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑦) → (V × {𝑥}) = (V × {(𝐺‘𝑦)}))
17	14, 16	eqtrid 2777	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑦) → (◡(2nd ↾ (V × V)) “ {𝑥}) = (V × {(𝐺‘𝑦)}))
18	15	imaeq2d 6034	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑦) → (({(𝐺‘𝑦)} × (V × {𝑦})) “ {𝑥}) = (({(𝐺‘𝑦)} × (V × {𝑦})) “ {(𝐺‘𝑦)}))
19		df-ima 5654	. . . . . . . . . . 11 ⊢ (({(𝐺‘𝑦)} × (V × {𝑦})) “ {(𝐺‘𝑦)}) = ran (({(𝐺‘𝑦)} × (V × {𝑦})) ↾ {(𝐺‘𝑦)})
20		ssid 3972	. . . . . . . . . . . . . 14 ⊢ {(𝐺‘𝑦)} ⊆ {(𝐺‘𝑦)}
21		xpssres 5992	. . . . . . . . . . . . . 14 ⊢ ({(𝐺‘𝑦)} ⊆ {(𝐺‘𝑦)} → (({(𝐺‘𝑦)} × (V × {𝑦})) ↾ {(𝐺‘𝑦)}) = ({(𝐺‘𝑦)} × (V × {𝑦})))
22	20, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (({(𝐺‘𝑦)} × (V × {𝑦})) ↾ {(𝐺‘𝑦)}) = ({(𝐺‘𝑦)} × (V × {𝑦}))
23	22	rneqi 5904	. . . . . . . . . . . 12 ⊢ ran (({(𝐺‘𝑦)} × (V × {𝑦})) ↾ {(𝐺‘𝑦)}) = ran ({(𝐺‘𝑦)} × (V × {𝑦}))
24	13	snnz 4743	. . . . . . . . . . . . 13 ⊢ {(𝐺‘𝑦)} ≠ ∅
25		rnxp 6146	. . . . . . . . . . . . 13 ⊢ ({(𝐺‘𝑦)} ≠ ∅ → ran ({(𝐺‘𝑦)} × (V × {𝑦})) = (V × {𝑦}))
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran ({(𝐺‘𝑦)} × (V × {𝑦})) = (V × {𝑦})
27	23, 26	eqtri 2753	. . . . . . . . . . 11 ⊢ ran (({(𝐺‘𝑦)} × (V × {𝑦})) ↾ {(𝐺‘𝑦)}) = (V × {𝑦})
28	19, 27	eqtri 2753	. . . . . . . . . 10 ⊢ (({(𝐺‘𝑦)} × (V × {𝑦})) “ {(𝐺‘𝑦)}) = (V × {𝑦})
29	18, 28	eqtrdi 2781	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑦) → (({(𝐺‘𝑦)} × (V × {𝑦})) “ {𝑥}) = (V × {𝑦}))
30	17, 29	xpeq12d 5672	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑦) → ((◡(2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺‘𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺‘𝑦)}) × (V × {𝑦})))
31	13, 30	iunxsn 5058	. . . . . . 7 ⊢ ∪ 𝑥 ∈ {(𝐺‘𝑦)} ((◡(2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺‘𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺‘𝑦)}) × (V × {𝑦}))
32	12, 31	eqtri 2753	. . . . . 6 ⊢ (({(𝐺‘𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺‘𝑦)}) × (V × {𝑦}))
33	32	cnveqi 5841	. . . . 5 ⊢ ◡(({(𝐺‘𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ◡((V × {(𝐺‘𝑦)}) × (V × {𝑦}))
34		cnvco 5852	. . . . 5 ⊢ ◡(({(𝐺‘𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ ◡({(𝐺‘𝑦)} × (V × {𝑦})))
35		cnvxp 6133	. . . . 5 ⊢ ◡((V × {(𝐺‘𝑦)}) × (V × {𝑦})) = ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))
36	33, 34, 35	3eqtr3i 2761	. . . 4 ⊢ (◡(2nd ↾ (V × V)) ∘ ◡({(𝐺‘𝑦)} × (V × {𝑦}))) = ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))
37		fparlem2 8095	. . . . . . . . 9 ⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
38	37	xpeq2i 5668	. . . . . . . 8 ⊢ ({(𝐺‘𝑦)} × (◡(2nd ↾ (V × V)) “ {𝑦})) = ({(𝐺‘𝑦)} × (V × {𝑦}))
39		fnsnfv 6943	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → {(𝐺‘𝑦)} = (𝐺 “ {𝑦}))
40	39	xpeq1d 5670	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → ({(𝐺‘𝑦)} × (◡(2nd ↾ (V × V)) “ {𝑦})) = ((𝐺 “ {𝑦}) × (◡(2nd ↾ (V × V)) “ {𝑦})))
41	38, 40	eqtr3id 2779	. . . . . . 7 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → ({(𝐺‘𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × (◡(2nd ↾ (V × V)) “ {𝑦})))
42	41	cnveqd 5842	. . . . . 6 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → ◡({(𝐺‘𝑦)} × (V × {𝑦})) = ◡((𝐺 “ {𝑦}) × (◡(2nd ↾ (V × V)) “ {𝑦})))
43		cnvxp 6133	. . . . . 6 ⊢ ◡((𝐺 “ {𝑦}) × (◡(2nd ↾ (V × V)) “ {𝑦})) = ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))
44	42, 43	eqtrdi 2781	. . . . 5 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → ◡({(𝐺‘𝑦)} × (V × {𝑦})) = ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
45	44	coeq2d 5829	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → (◡(2nd ↾ (V × V)) ∘ ◡({(𝐺‘𝑦)} × (V × {𝑦}))) = (◡(2nd ↾ (V × V)) ∘ ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
46	36, 45	eqtr3id 2779	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → ((V × {𝑦}) × (V × {(𝐺‘𝑦)})) = (◡(2nd ↾ (V × V)) ∘ ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
47	46	iuneq2dv 4983	. 2 ⊢ (𝐺 Fn 𝐵 → ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)})) = ∪ 𝑦 ∈ 𝐵 (◡(2nd ↾ (V × V)) ∘ ((◡(2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
48	1, 7, 47	3eqtr4a 2791	1 ⊢ (𝐺 Fn 𝐵 → (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  {csn 4592  ∪ ciun 4958   × cxp 5639  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   ↾ cres 5643   “ cima 5644   ∘ ccom 5645   Fn wfn 6509  ‘cfv 6514  2^nd c2nd 7970

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-1st 7971  df-2nd 7972

This theorem is referenced by:  fpar  8098