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Theorem fparlem4 7955
Description: Lemma for fpar 7956. (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)))) = 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
Distinct variable groups:   𝑦,𝐵   𝑦,𝐺

Proof of Theorem fparlem4
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 coiun 6160 . 2 ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
2 inss1 4162 . . . . 5 (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ dom 𝐺
3 fndm 6536 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3sseqtrid 3973 . . . 4 (𝐺 Fn 𝐵 → (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ 𝐵)
5 dfco2a 6150 . . . 4 ((dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ 𝐵 → (𝐺 ∘ (2nd ↾ (V × V))) = 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
64, 5syl 17 . . 3 (𝐺 Fn 𝐵 → (𝐺 ∘ (2nd ↾ (V × V))) = 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
76coeq2d 5771 . 2 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
8 inss1 4162 . . . . . . . . 9 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ dom ({(𝐺𝑦)} × (V × {𝑦}))
9 dmxpss 6074 . . . . . . . . 9 dom ({(𝐺𝑦)} × (V × {𝑦})) ⊆ {(𝐺𝑦)}
108, 9sstri 3930 . . . . . . . 8 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ {(𝐺𝑦)}
11 dfco2a 6150 . . . . . . . 8 ((dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ {(𝐺𝑦)} → (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = 𝑥 ∈ {(𝐺𝑦)} (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})))
1210, 11ax-mp 5 . . . . . . 7 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = 𝑥 ∈ {(𝐺𝑦)} (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}))
13 fvex 6787 . . . . . . . 8 (𝐺𝑦) ∈ V
14 fparlem2 7953 . . . . . . . . . 10 ((2nd ↾ (V × V)) “ {𝑥}) = (V × {𝑥})
15 sneq 4571 . . . . . . . . . . 11 (𝑥 = (𝐺𝑦) → {𝑥} = {(𝐺𝑦)})
1615xpeq2d 5619 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (V × {𝑥}) = (V × {(𝐺𝑦)}))
1714, 16eqtrid 2790 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → ((2nd ↾ (V × V)) “ {𝑥}) = (V × {(𝐺𝑦)}))
1815imaeq2d 5969 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}))
19 df-ima 5602 . . . . . . . . . . 11 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)})
20 ssid 3943 . . . . . . . . . . . . . 14 {(𝐺𝑦)} ⊆ {(𝐺𝑦)}
21 xpssres 5928 . . . . . . . . . . . . . 14 ({(𝐺𝑦)} ⊆ {(𝐺𝑦)} → (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦})))
2220, 21ax-mp 5 . . . . . . . . . . . . 13 (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦}))
2322rneqi 5846 . . . . . . . . . . . 12 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ran ({(𝐺𝑦)} × (V × {𝑦}))
2413snnz 4712 . . . . . . . . . . . . 13 {(𝐺𝑦)} ≠ ∅
25 rnxp 6073 . . . . . . . . . . . . 13 ({(𝐺𝑦)} ≠ ∅ → ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦}))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦})
2723, 26eqtri 2766 . . . . . . . . . . 11 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = (V × {𝑦})
2819, 27eqtri 2766 . . . . . . . . . 10 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = (V × {𝑦})
2918, 28eqtrdi 2794 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (V × {𝑦}))
3017, 29xpeq12d 5620 . . . . . . . 8 (𝑥 = (𝐺𝑦) → (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦})))
3113, 30iunxsn 5020 . . . . . . 7 𝑥 ∈ {(𝐺𝑦)} (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3212, 31eqtri 2766 . . . . . 6 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3332cnveqi 5783 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
34 cnvco 5794 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦})))
35 cnvxp 6060 . . . . 5 ((V × {(𝐺𝑦)}) × (V × {𝑦})) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
3633, 34, 353eqtr3i 2774 . . . 4 ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
37 fparlem2 7953 . . . . . . . . 9 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
3837xpeq2i 5616 . . . . . . . 8 ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ({(𝐺𝑦)} × (V × {𝑦}))
39 fnsnfv 6847 . . . . . . . . 9 ((𝐺 Fn 𝐵𝑦𝐵) → {(𝐺𝑦)} = (𝐺 “ {𝑦}))
4039xpeq1d 5618 . . . . . . . 8 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4138, 40eqtr3id 2792 . . . . . . 7 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4241cnveqd 5784 . . . . . 6 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
43 cnvxp 6060 . . . . . 6 ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))
4442, 43eqtrdi 2794 . . . . 5 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
4544coeq2d 5771 . . . 4 ((𝐺 Fn 𝐵𝑦𝐵) → ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4636, 45eqtr3id 2792 . . 3 ((𝐺 Fn 𝐵𝑦𝐵) → ((V × {𝑦}) × (V × {(𝐺𝑦)})) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4746iuneq2dv 4948 . 2 (𝐺 Fn 𝐵 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
481, 7, 473eqtr4a 2804 1 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wne 2943  Vcvv 3432  cin 3886  wss 3887  c0 4256  {csn 4561   ciun 4924   × cxp 5587  ccnv 5588  dom cdm 5589  ran crn 5590  cres 5591  cima 5592  ccom 5593   Fn wfn 6428  cfv 6433  2nd c2nd 7830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-1st 7831  df-2nd 7832
This theorem is referenced by:  fpar  7956
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