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Theorem fparlem4 7226
Description: Lemma for fpar 7227. (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 5607 . 2 ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
2 inss1 3816 . . . . 5 (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ dom 𝐺
3 fndm 5950 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3syl5sseq 3637 . . . 4 (𝐺 Fn 𝐵 → (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ 𝐵)
5 dfco2a 5597 . . . 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 5249 . 2 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
8 inss1 3816 . . . . . . . . 9 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ dom ({(𝐺𝑦)} × (V × {𝑦}))
9 dmxpss 5528 . . . . . . . . 9 dom ({(𝐺𝑦)} × (V × {𝑦})) ⊆ {(𝐺𝑦)}
108, 9sstri 3597 . . . . . . . 8 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ {(𝐺𝑦)}
11 dfco2a 5597 . . . . . . . 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 6160 . . . . . . . 8 (𝐺𝑦) ∈ V
14 fparlem2 7224 . . . . . . . . . 10 ((2nd ↾ (V × V)) “ {𝑥}) = (V × {𝑥})
15 sneq 4163 . . . . . . . . . . 11 (𝑥 = (𝐺𝑦) → {𝑥} = {(𝐺𝑦)})
1615xpeq2d 5104 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (V × {𝑥}) = (V × {(𝐺𝑦)}))
1714, 16syl5eq 2672 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → ((2nd ↾ (V × V)) “ {𝑥}) = (V × {(𝐺𝑦)}))
1815imaeq2d 5429 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}))
19 df-ima 5092 . . . . . . . . . . 11 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)})
20 ssid 3608 . . . . . . . . . . . . . 14 {(𝐺𝑦)} ⊆ {(𝐺𝑦)}
21 xpssres 5397 . . . . . . . . . . . . . 14 ({(𝐺𝑦)} ⊆ {(𝐺𝑦)} → (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦})))
2220, 21ax-mp 5 . . . . . . . . . . . . 13 (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦}))
2322rneqi 5316 . . . . . . . . . . . 12 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ran ({(𝐺𝑦)} × (V × {𝑦}))
2413snnz 4284 . . . . . . . . . . . . 13 {(𝐺𝑦)} ≠ ∅
25 rnxp 5527 . . . . . . . . . . . . 13 ({(𝐺𝑦)} ≠ ∅ → ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦}))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦})
2723, 26eqtri 2648 . . . . . . . . . . 11 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = (V × {𝑦})
2819, 27eqtri 2648 . . . . . . . . . 10 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = (V × {𝑦})
2918, 28syl6eq 2676 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (V × {𝑦}))
3017, 29xpeq12d 5105 . . . . . . . 8 (𝑥 = (𝐺𝑦) → (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦})))
3113, 30iunxsn 4574 . . . . . . 7 𝑥 ∈ {(𝐺𝑦)} (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3212, 31eqtri 2648 . . . . . 6 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3332cnveqi 5262 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
34 cnvco 5273 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦})))
35 cnvxp 5514 . . . . 5 ((V × {(𝐺𝑦)}) × (V × {𝑦})) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
3633, 34, 353eqtr3i 2656 . . . 4 ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
37 fparlem2 7224 . . . . . . . . 9 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
3837xpeq2i 5101 . . . . . . . 8 ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ({(𝐺𝑦)} × (V × {𝑦}))
39 fnsnfv 6216 . . . . . . . . 9 ((𝐺 Fn 𝐵𝑦𝐵) → {(𝐺𝑦)} = (𝐺 “ {𝑦}))
4039xpeq1d 5103 . . . . . . . 8 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4138, 40syl5eqr 2674 . . . . . . 7 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4241cnveqd 5263 . . . . . 6 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
43 cnvxp 5514 . . . . . 6 ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))
4442, 43syl6eq 2676 . . . . 5 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
4544coeq2d 5249 . . . 4 ((𝐺 Fn 𝐵𝑦𝐵) → ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4636, 45syl5eqr 2674 . . 3 ((𝐺 Fn 𝐵𝑦𝐵) → ((V × {𝑦}) × (V × {(𝐺𝑦)})) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4746iuneq2dv 4513 . 2 (𝐺 Fn 𝐵 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
481, 7, 473eqtr4a 2686 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 384   = wceq 1480  wcel 1992  wne 2796  Vcvv 3191  cin 3559  wss 3560  c0 3896  {csn 4153   ciun 4490   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  ccom 5083   Fn wfn 5845  cfv 5850  2nd c2nd 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-1st 7116  df-2nd 7117
This theorem is referenced by:  fpar  7227
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