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Theorem fparlem4 7799
Description: Lemma for fpar 7800. (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 6102 . 2 ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
2 inss1 4202 . . . . 5 (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ dom 𝐺
3 fndm 6448 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3sseqtrid 4016 . . . 4 (𝐺 Fn 𝐵 → (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ 𝐵)
5 dfco2a 6092 . . . 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 5726 . 2 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
8 inss1 4202 . . . . . . . . 9 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ dom ({(𝐺𝑦)} × (V × {𝑦}))
9 dmxpss 6021 . . . . . . . . 9 dom ({(𝐺𝑦)} × (V × {𝑦})) ⊆ {(𝐺𝑦)}
108, 9sstri 3973 . . . . . . . 8 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ {(𝐺𝑦)}
11 dfco2a 6092 . . . . . . . 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 6676 . . . . . . . 8 (𝐺𝑦) ∈ V
14 fparlem2 7797 . . . . . . . . . 10 ((2nd ↾ (V × V)) “ {𝑥}) = (V × {𝑥})
15 sneq 4567 . . . . . . . . . . 11 (𝑥 = (𝐺𝑦) → {𝑥} = {(𝐺𝑦)})
1615xpeq2d 5578 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (V × {𝑥}) = (V × {(𝐺𝑦)}))
1714, 16syl5eq 2865 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → ((2nd ↾ (V × V)) “ {𝑥}) = (V × {(𝐺𝑦)}))
1815imaeq2d 5922 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}))
19 df-ima 5561 . . . . . . . . . . 11 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)})
20 ssid 3986 . . . . . . . . . . . . . 14 {(𝐺𝑦)} ⊆ {(𝐺𝑦)}
21 xpssres 5882 . . . . . . . . . . . . . 14 ({(𝐺𝑦)} ⊆ {(𝐺𝑦)} → (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦})))
2220, 21ax-mp 5 . . . . . . . . . . . . 13 (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦}))
2322rneqi 5800 . . . . . . . . . . . 12 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ran ({(𝐺𝑦)} × (V × {𝑦}))
2413snnz 4703 . . . . . . . . . . . . 13 {(𝐺𝑦)} ≠ ∅
25 rnxp 6020 . . . . . . . . . . . . 13 ({(𝐺𝑦)} ≠ ∅ → ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦}))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦})
2723, 26eqtri 2841 . . . . . . . . . . 11 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = (V × {𝑦})
2819, 27eqtri 2841 . . . . . . . . . 10 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = (V × {𝑦})
2918, 28syl6eq 2869 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (V × {𝑦}))
3017, 29xpeq12d 5579 . . . . . . . 8 (𝑥 = (𝐺𝑦) → (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦})))
3113, 30iunxsn 5004 . . . . . . 7 𝑥 ∈ {(𝐺𝑦)} (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3212, 31eqtri 2841 . . . . . 6 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3332cnveqi 5738 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
34 cnvco 5749 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦})))
35 cnvxp 6007 . . . . 5 ((V × {(𝐺𝑦)}) × (V × {𝑦})) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
3633, 34, 353eqtr3i 2849 . . . 4 ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
37 fparlem2 7797 . . . . . . . . 9 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
3837xpeq2i 5575 . . . . . . . 8 ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ({(𝐺𝑦)} × (V × {𝑦}))
39 fnsnfv 6736 . . . . . . . . 9 ((𝐺 Fn 𝐵𝑦𝐵) → {(𝐺𝑦)} = (𝐺 “ {𝑦}))
4039xpeq1d 5577 . . . . . . . 8 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4138, 40syl5eqr 2867 . . . . . . 7 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4241cnveqd 5739 . . . . . 6 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
43 cnvxp 6007 . . . . . 6 ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))
4442, 43syl6eq 2869 . . . . 5 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
4544coeq2d 5726 . . . 4 ((𝐺 Fn 𝐵𝑦𝐵) → ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4636, 45syl5eqr 2867 . . 3 ((𝐺 Fn 𝐵𝑦𝐵) → ((V × {𝑦}) × (V × {(𝐺𝑦)})) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4746iuneq2dv 4934 . 2 (𝐺 Fn 𝐵 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
481, 7, 473eqtr4a 2879 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 1528  wcel 2105  wne 3013  Vcvv 3492  cin 3932  wss 3933  c0 4288  {csn 4557   ciun 4910   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  ccom 5552   Fn wfn 6343  cfv 6348  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-1st 7678  df-2nd 7679
This theorem is referenced by:  fpar  7800
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