MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fparlem3 Structured version   Visualization version   GIF version

Theorem fparlem3 7225
Description: Lemma for fpar 7227. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fparlem3 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem fparlem3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 coiun 5607 . 2 ((1st ↾ (V × V)) ∘ 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))) = 𝑥𝐴 ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
2 inss1 3816 . . . . 5 (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ dom 𝐹
3 fndm 5950 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
42, 3syl5sseq 3637 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ 𝐴)
5 dfco2a 5597 . . . 4 ((dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ 𝐴 → (𝐹 ∘ (1st ↾ (V × V))) = 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
64, 5syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∘ (1st ↾ (V × V))) = 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
76coeq2d 5249 . 2 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ((1st ↾ (V × V)) ∘ 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8 inss1 3816 . . . . . . . . 9 (dom ({(𝐹𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ dom ({(𝐹𝑥)} × ({𝑥} × V))
9 dmxpss 5528 . . . . . . . . 9 dom ({(𝐹𝑥)} × ({𝑥} × V)) ⊆ {(𝐹𝑥)}
108, 9sstri 3597 . . . . . . . 8 (dom ({(𝐹𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ {(𝐹𝑥)}
11 dfco2a 5597 . . . . . . . 8 ((dom ({(𝐹𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ {(𝐹𝑥)} → (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = 𝑦 ∈ {(𝐹𝑥)} (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦})))
1210, 11ax-mp 5 . . . . . . 7 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = 𝑦 ∈ {(𝐹𝑥)} (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦}))
13 fvex 6160 . . . . . . . 8 (𝐹𝑥) ∈ V
14 fparlem1 7223 . . . . . . . . . 10 ((1st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15 sneq 4163 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) → {𝑦} = {(𝐹𝑥)})
1615xpeq1d 5103 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ({𝑦} × V) = ({(𝐹𝑥)} × V))
1714, 16syl5eq 2672 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((1st ↾ (V × V)) “ {𝑦}) = ({(𝐹𝑥)} × V))
1815imaeq2d 5429 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}))
19 df-ima 5092 . . . . . . . . . . 11 (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}) = ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)})
20 ssid 3608 . . . . . . . . . . . . . 14 {(𝐹𝑥)} ⊆ {(𝐹𝑥)}
21 xpssres 5397 . . . . . . . . . . . . . 14 ({(𝐹𝑥)} ⊆ {(𝐹𝑥)} → (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({(𝐹𝑥)} × ({𝑥} × V)))
2220, 21ax-mp 5 . . . . . . . . . . . . 13 (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({(𝐹𝑥)} × ({𝑥} × V))
2322rneqi 5316 . . . . . . . . . . . 12 ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ran ({(𝐹𝑥)} × ({𝑥} × V))
2413snnz 4284 . . . . . . . . . . . . 13 {(𝐹𝑥)} ≠ ∅
25 rnxp 5527 . . . . . . . . . . . . 13 ({(𝐹𝑥)} ≠ ∅ → ran ({(𝐹𝑥)} × ({𝑥} × V)) = ({𝑥} × V))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ran ({(𝐹𝑥)} × ({𝑥} × V)) = ({𝑥} × V)
2723, 26eqtri 2648 . . . . . . . . . . 11 ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({𝑥} × V)
2819, 27eqtri 2648 . . . . . . . . . 10 (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}) = ({𝑥} × V)
2918, 28syl6eq 2676 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
3017, 29xpeq12d 5105 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹𝑥)} × V) × ({𝑥} × V)))
3113, 30iunxsn 4574 . . . . . . 7 𝑦 ∈ {(𝐹𝑥)} (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
3212, 31eqtri 2648 . . . . . 6 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
3332cnveqi 5262 . . . . 5 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
34 cnvco 5273 . . . . 5 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V)))
35 cnvxp 5514 . . . . 5 (({(𝐹𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹𝑥)} × V))
3633, 34, 353eqtr3i 2656 . . . 4 ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹𝑥)} × V))
37 fparlem1 7223 . . . . . . . . 9 ((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
3837xpeq2i 5101 . . . . . . . 8 ({(𝐹𝑥)} × ((1st ↾ (V × V)) “ {𝑥})) = ({(𝐹𝑥)} × ({𝑥} × V))
39 fnsnfv 6216 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → {(𝐹𝑥)} = (𝐹 “ {𝑥}))
4039xpeq1d 5103 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ((1st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
4138, 40syl5eqr 2674 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
4241cnveqd 5263 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
43 cnvxp 5514 . . . . . 6 ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})) = (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
4442, 43syl6eq 2676 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
4544coeq2d 5249 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V))) = ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
4636, 45syl5eqr 2674 . . 3 ((𝐹 Fn 𝐴𝑥𝐴) → (({𝑥} × V) × ({(𝐹𝑥)} × V)) = ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
4746iuneq2dv 4513 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) = 𝑥𝐴 ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
481, 7, 473eqtr4a 2686 1 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)))
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  1st c1st 7114
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
  Copyright terms: Public domain W3C validator