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Theorem fparlem3 7435
Description: Lemma for fpar 7437. (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 5794 . 2 ((1st ↾ (V × V)) ∘ 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))) = 𝑥𝐴 ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
2 inss1 3964 . . . . 5 (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ dom 𝐹
3 fndm 6139 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
42, 3syl5sseq 3782 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ 𝐴)
5 dfco2a 5784 . . . 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 5428 . 2 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ((1st ↾ (V × V)) ∘ 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8 inss1 3964 . . . . . . . . 9 (dom ({(𝐹𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ dom ({(𝐹𝑥)} × ({𝑥} × V))
9 dmxpss 5711 . . . . . . . . 9 dom ({(𝐹𝑥)} × ({𝑥} × V)) ⊆ {(𝐹𝑥)}
108, 9sstri 3741 . . . . . . . 8 (dom ({(𝐹𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ {(𝐹𝑥)}
11 dfco2a 5784 . . . . . . . 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 6350 . . . . . . . 8 (𝐹𝑥) ∈ V
14 fparlem1 7433 . . . . . . . . . 10 ((1st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15 sneq 4319 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) → {𝑦} = {(𝐹𝑥)})
1615xpeq1d 5283 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ({𝑦} × V) = ({(𝐹𝑥)} × V))
1714, 16syl5eq 2794 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((1st ↾ (V × V)) “ {𝑦}) = ({(𝐹𝑥)} × V))
1815imaeq2d 5612 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}))
19 df-ima 5267 . . . . . . . . . . 11 (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}) = ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)})
20 ssid 3753 . . . . . . . . . . . . . 14 {(𝐹𝑥)} ⊆ {(𝐹𝑥)}
21 xpssres 5580 . . . . . . . . . . . . . 14 ({(𝐹𝑥)} ⊆ {(𝐹𝑥)} → (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({(𝐹𝑥)} × ({𝑥} × V)))
2220, 21ax-mp 5 . . . . . . . . . . . . 13 (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({(𝐹𝑥)} × ({𝑥} × V))
2322rneqi 5495 . . . . . . . . . . . 12 ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ran ({(𝐹𝑥)} × ({𝑥} × V))
2413snnz 4440 . . . . . . . . . . . . 13 {(𝐹𝑥)} ≠ ∅
25 rnxp 5710 . . . . . . . . . . . . 13 ({(𝐹𝑥)} ≠ ∅ → ran ({(𝐹𝑥)} × ({𝑥} × V)) = ({𝑥} × V))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ran ({(𝐹𝑥)} × ({𝑥} × V)) = ({𝑥} × V)
2723, 26eqtri 2770 . . . . . . . . . . 11 ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({𝑥} × V)
2819, 27eqtri 2770 . . . . . . . . . 10 (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}) = ({𝑥} × V)
2918, 28syl6eq 2798 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
3017, 29xpeq12d 5285 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹𝑥)} × V) × ({𝑥} × V)))
3113, 30iunxsn 4743 . . . . . . 7 𝑦 ∈ {(𝐹𝑥)} (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
3212, 31eqtri 2770 . . . . . 6 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
3332cnveqi 5440 . . . . 5 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
34 cnvco 5451 . . . . 5 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V)))
35 cnvxp 5697 . . . . 5 (({(𝐹𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹𝑥)} × V))
3633, 34, 353eqtr3i 2778 . . . 4 ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹𝑥)} × V))
37 fparlem1 7433 . . . . . . . . 9 ((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
3837xpeq2i 5281 . . . . . . . 8 ({(𝐹𝑥)} × ((1st ↾ (V × V)) “ {𝑥})) = ({(𝐹𝑥)} × ({𝑥} × V))
39 fnsnfv 6408 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → {(𝐹𝑥)} = (𝐹 “ {𝑥}))
4039xpeq1d 5283 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ((1st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
4138, 40syl5eqr 2796 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
4241cnveqd 5441 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
43 cnvxp 5697 . . . . . 6 ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})) = (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
4442, 43syl6eq 2798 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
4544coeq2d 5428 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V))) = ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
4636, 45syl5eqr 2796 . . 3 ((𝐹 Fn 𝐴𝑥𝐴) → (({𝑥} × V) × ({(𝐹𝑥)} × V)) = ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
4746iuneq2dv 4682 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) = 𝑥𝐴 ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
481, 7, 473eqtr4a 2808 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 383   = wceq 1620  wcel 2127  wne 2920  Vcvv 3328  cin 3702  wss 3703  c0 4046  {csn 4309   ciun 4660   × cxp 5252  ccnv 5253  dom cdm 5254  ran crn 5255  cres 5256  cima 5257  ccom 5258   Fn wfn 6032  cfv 6037  1st c1st 7319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-1st 7321  df-2nd 7322
This theorem is referenced by:  fpar  7437
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