MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fparlem4 Structured version   Visualization version   GIF version

Theorem fparlem4 7448
Description: Lemma for fpar 7449. (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 5806 . 2 ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
2 inss1 3976 . . . . 5 (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ dom 𝐺
3 fndm 6151 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3syl5sseq 3794 . . . 4 (𝐺 Fn 𝐵 → (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ 𝐵)
5 dfco2a 5796 . . . 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 5440 . 2 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
8 inss1 3976 . . . . . . . . 9 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ dom ({(𝐺𝑦)} × (V × {𝑦}))
9 dmxpss 5723 . . . . . . . . 9 dom ({(𝐺𝑦)} × (V × {𝑦})) ⊆ {(𝐺𝑦)}
108, 9sstri 3753 . . . . . . . 8 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ {(𝐺𝑦)}
11 dfco2a 5796 . . . . . . . 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 6362 . . . . . . . 8 (𝐺𝑦) ∈ V
14 fparlem2 7446 . . . . . . . . . 10 ((2nd ↾ (V × V)) “ {𝑥}) = (V × {𝑥})
15 sneq 4331 . . . . . . . . . . 11 (𝑥 = (𝐺𝑦) → {𝑥} = {(𝐺𝑦)})
1615xpeq2d 5296 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (V × {𝑥}) = (V × {(𝐺𝑦)}))
1714, 16syl5eq 2806 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → ((2nd ↾ (V × V)) “ {𝑥}) = (V × {(𝐺𝑦)}))
1815imaeq2d 5624 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}))
19 df-ima 5279 . . . . . . . . . . 11 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)})
20 ssid 3765 . . . . . . . . . . . . . 14 {(𝐺𝑦)} ⊆ {(𝐺𝑦)}
21 xpssres 5592 . . . . . . . . . . . . . 14 ({(𝐺𝑦)} ⊆ {(𝐺𝑦)} → (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦})))
2220, 21ax-mp 5 . . . . . . . . . . . . 13 (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦}))
2322rneqi 5507 . . . . . . . . . . . 12 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ran ({(𝐺𝑦)} × (V × {𝑦}))
2413snnz 4452 . . . . . . . . . . . . 13 {(𝐺𝑦)} ≠ ∅
25 rnxp 5722 . . . . . . . . . . . . 13 ({(𝐺𝑦)} ≠ ∅ → ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦}))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦})
2723, 26eqtri 2782 . . . . . . . . . . 11 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = (V × {𝑦})
2819, 27eqtri 2782 . . . . . . . . . 10 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = (V × {𝑦})
2918, 28syl6eq 2810 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (V × {𝑦}))
3017, 29xpeq12d 5297 . . . . . . . 8 (𝑥 = (𝐺𝑦) → (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦})))
3113, 30iunxsn 4755 . . . . . . 7 𝑥 ∈ {(𝐺𝑦)} (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3212, 31eqtri 2782 . . . . . 6 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3332cnveqi 5452 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
34 cnvco 5463 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦})))
35 cnvxp 5709 . . . . 5 ((V × {(𝐺𝑦)}) × (V × {𝑦})) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
3633, 34, 353eqtr3i 2790 . . . 4 ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
37 fparlem2 7446 . . . . . . . . 9 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
3837xpeq2i 5293 . . . . . . . 8 ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ({(𝐺𝑦)} × (V × {𝑦}))
39 fnsnfv 6420 . . . . . . . . 9 ((𝐺 Fn 𝐵𝑦𝐵) → {(𝐺𝑦)} = (𝐺 “ {𝑦}))
4039xpeq1d 5295 . . . . . . . 8 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4138, 40syl5eqr 2808 . . . . . . 7 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4241cnveqd 5453 . . . . . 6 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
43 cnvxp 5709 . . . . . 6 ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))
4442, 43syl6eq 2810 . . . . 5 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
4544coeq2d 5440 . . . 4 ((𝐺 Fn 𝐵𝑦𝐵) → ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4636, 45syl5eqr 2808 . . 3 ((𝐺 Fn 𝐵𝑦𝐵) → ((V × {𝑦}) × (V × {(𝐺𝑦)})) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4746iuneq2dv 4694 . 2 (𝐺 Fn 𝐵 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
481, 7, 473eqtr4a 2820 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 383   = wceq 1632  wcel 2139  wne 2932  Vcvv 3340  cin 3714  wss 3715  c0 4058  {csn 4321   ciun 4672   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  cres 5268  cima 5269  ccom 5270   Fn wfn 6044  cfv 6049  2nd c2nd 7332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-1st 7333  df-2nd 7334
This theorem is referenced by:  fpar  7449
  Copyright terms: Public domain W3C validator