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Theorem fparlem3 8102
Description: Lemma for fpar 8104. (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 6255 . 2 ((1st ↾ (V × V)) ∘ 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))) = 𝑥𝐴 ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
2 inss1 4228 . . . . 5 (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ dom 𝐹
3 fndm 6652 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
42, 3sseqtrid 4034 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ 𝐴)
5 dfco2a 6245 . . . 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 5862 . 2 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ((1st ↾ (V × V)) ∘ 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8 inss1 4228 . . . . . . . . 9 (dom ({(𝐹𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ dom ({(𝐹𝑥)} × ({𝑥} × V))
9 dmxpss 6170 . . . . . . . . 9 dom ({(𝐹𝑥)} × ({𝑥} × V)) ⊆ {(𝐹𝑥)}
108, 9sstri 3991 . . . . . . . 8 (dom ({(𝐹𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ {(𝐹𝑥)}
11 dfco2a 6245 . . . . . . . 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 6904 . . . . . . . 8 (𝐹𝑥) ∈ V
14 fparlem1 8100 . . . . . . . . . 10 ((1st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15 sneq 4638 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) → {𝑦} = {(𝐹𝑥)})
1615xpeq1d 5705 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ({𝑦} × V) = ({(𝐹𝑥)} × V))
1714, 16eqtrid 2784 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((1st ↾ (V × V)) “ {𝑦}) = ({(𝐹𝑥)} × V))
1815imaeq2d 6059 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}))
19 df-ima 5689 . . . . . . . . . . 11 (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}) = ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)})
20 ssid 4004 . . . . . . . . . . . . . 14 {(𝐹𝑥)} ⊆ {(𝐹𝑥)}
21 xpssres 6018 . . . . . . . . . . . . . 14 ({(𝐹𝑥)} ⊆ {(𝐹𝑥)} → (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({(𝐹𝑥)} × ({𝑥} × V)))
2220, 21ax-mp 5 . . . . . . . . . . . . 13 (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({(𝐹𝑥)} × ({𝑥} × V))
2322rneqi 5936 . . . . . . . . . . . 12 ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ran ({(𝐹𝑥)} × ({𝑥} × V))
2413snnz 4780 . . . . . . . . . . . . 13 {(𝐹𝑥)} ≠ ∅
25 rnxp 6169 . . . . . . . . . . . . 13 ({(𝐹𝑥)} ≠ ∅ → ran ({(𝐹𝑥)} × ({𝑥} × V)) = ({𝑥} × V))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ran ({(𝐹𝑥)} × ({𝑥} × V)) = ({𝑥} × V)
2723, 26eqtri 2760 . . . . . . . . . . 11 ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({𝑥} × V)
2819, 27eqtri 2760 . . . . . . . . . 10 (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}) = ({𝑥} × V)
2918, 28eqtrdi 2788 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
3017, 29xpeq12d 5707 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹𝑥)} × V) × ({𝑥} × V)))
3113, 30iunxsn 5094 . . . . . . 7 𝑦 ∈ {(𝐹𝑥)} (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
3212, 31eqtri 2760 . . . . . 6 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
3332cnveqi 5874 . . . . 5 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
34 cnvco 5885 . . . . 5 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V)))
35 cnvxp 6156 . . . . 5 (({(𝐹𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹𝑥)} × V))
3633, 34, 353eqtr3i 2768 . . . 4 ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹𝑥)} × V))
37 fparlem1 8100 . . . . . . . . 9 ((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
3837xpeq2i 5703 . . . . . . . 8 ({(𝐹𝑥)} × ((1st ↾ (V × V)) “ {𝑥})) = ({(𝐹𝑥)} × ({𝑥} × V))
39 fnsnfv 6970 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → {(𝐹𝑥)} = (𝐹 “ {𝑥}))
4039xpeq1d 5705 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ((1st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
4138, 40eqtr3id 2786 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
4241cnveqd 5875 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
43 cnvxp 6156 . . . . . 6 ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})) = (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
4442, 43eqtrdi 2788 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
4544coeq2d 5862 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V))) = ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
4636, 45eqtr3id 2786 . . 3 ((𝐹 Fn 𝐴𝑥𝐴) → (({𝑥} × V) × ({(𝐹𝑥)} × V)) = ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
4746iuneq2dv 5021 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) = 𝑥𝐴 ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
481, 7, 473eqtr4a 2798 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 396   = wceq 1541  wcel 2106  wne 2940  Vcvv 3474  cin 3947  wss 3948  c0 4322  {csn 4628   ciun 4997   × cxp 5674  ccnv 5675  dom cdm 5676  ran crn 5677  cres 5678  cima 5679  ccom 5680   Fn wfn 6538  cfv 6543  1st c1st 7975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-1st 7977  df-2nd 7978
This theorem is referenced by:  fpar  8104
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