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Theorem fparlem3 8137
Description: Lemma for fpar 8139. (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 6277 . 2 ((1st ↾ (V × V)) ∘ 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))) = 𝑥𝐴 ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
2 inss1 4244 . . . . 5 (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ dom 𝐹
3 fndm 6671 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
42, 3sseqtrid 4047 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1st ↾ (V × V))) ⊆ 𝐴)
5 dfco2a 6267 . . . 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 5875 . 2 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ((1st ↾ (V × V)) ∘ 𝑥𝐴 (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8 inss1 4244 . . . . . . . . 9 (dom ({(𝐹𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ dom ({(𝐹𝑥)} × ({𝑥} × V))
9 dmxpss 6192 . . . . . . . . 9 dom ({(𝐹𝑥)} × ({𝑥} × V)) ⊆ {(𝐹𝑥)}
108, 9sstri 4004 . . . . . . . 8 (dom ({(𝐹𝑥)} × ({𝑥} × V)) ∩ ran (1st ↾ (V × V))) ⊆ {(𝐹𝑥)}
11 dfco2a 6267 . . . . . . . 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 6919 . . . . . . . 8 (𝐹𝑥) ∈ V
14 fparlem1 8135 . . . . . . . . . 10 ((1st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15 sneq 4640 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) → {𝑦} = {(𝐹𝑥)})
1615xpeq1d 5717 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ({𝑦} × V) = ({(𝐹𝑥)} × V))
1714, 16eqtrid 2786 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((1st ↾ (V × V)) “ {𝑦}) = ({(𝐹𝑥)} × V))
1815imaeq2d 6079 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}))
19 df-ima 5701 . . . . . . . . . . 11 (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}) = ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)})
20 ssid 4017 . . . . . . . . . . . . . 14 {(𝐹𝑥)} ⊆ {(𝐹𝑥)}
21 xpssres 6037 . . . . . . . . . . . . . 14 ({(𝐹𝑥)} ⊆ {(𝐹𝑥)} → (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({(𝐹𝑥)} × ({𝑥} × V)))
2220, 21ax-mp 5 . . . . . . . . . . . . 13 (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({(𝐹𝑥)} × ({𝑥} × V))
2322rneqi 5950 . . . . . . . . . . . 12 ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ran ({(𝐹𝑥)} × ({𝑥} × V))
2413snnz 4780 . . . . . . . . . . . . 13 {(𝐹𝑥)} ≠ ∅
25 rnxp 6191 . . . . . . . . . . . . 13 ({(𝐹𝑥)} ≠ ∅ → ran ({(𝐹𝑥)} × ({𝑥} × V)) = ({𝑥} × V))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ran ({(𝐹𝑥)} × ({𝑥} × V)) = ({𝑥} × V)
2723, 26eqtri 2762 . . . . . . . . . . 11 ran (({(𝐹𝑥)} × ({𝑥} × V)) ↾ {(𝐹𝑥)}) = ({𝑥} × V)
2819, 27eqtri 2762 . . . . . . . . . 10 (({(𝐹𝑥)} × ({𝑥} × V)) “ {(𝐹𝑥)}) = ({𝑥} × V)
2918, 28eqtrdi 2790 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
3017, 29xpeq12d 5719 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹𝑥)} × V) × ({𝑥} × V)))
3113, 30iunxsn 5095 . . . . . . 7 𝑦 ∈ {(𝐹𝑥)} (((1st ↾ (V × V)) “ {𝑦}) × (({(𝐹𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
3212, 31eqtri 2762 . . . . . 6 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
3332cnveqi 5887 . . . . 5 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = (({(𝐹𝑥)} × V) × ({𝑥} × V))
34 cnvco 5898 . . . . 5 (({(𝐹𝑥)} × ({𝑥} × V)) ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V)))
35 cnvxp 6178 . . . . 5 (({(𝐹𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹𝑥)} × V))
3633, 34, 353eqtr3i 2770 . . . 4 ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹𝑥)} × V))
37 fparlem1 8135 . . . . . . . . 9 ((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
3837xpeq2i 5715 . . . . . . . 8 ({(𝐹𝑥)} × ((1st ↾ (V × V)) “ {𝑥})) = ({(𝐹𝑥)} × ({𝑥} × V))
39 fnsnfv 6987 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → {(𝐹𝑥)} = (𝐹 “ {𝑥}))
4039xpeq1d 5717 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ((1st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
4138, 40eqtr3id 2788 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
4241cnveqd 5888 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})))
43 cnvxp 6178 . . . . . 6 ((𝐹 “ {𝑥}) × ((1st ↾ (V × V)) “ {𝑥})) = (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
4442, 43eqtrdi 2790 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ({(𝐹𝑥)} × ({𝑥} × V)) = (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
4544coeq2d 5875 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → ((1st ↾ (V × V)) ∘ ({(𝐹𝑥)} × ({𝑥} × V))) = ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
4636, 45eqtr3id 2788 . . 3 ((𝐹 Fn 𝐴𝑥𝐴) → (({𝑥} × V) × ({(𝐹𝑥)} × V)) = ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
4746iuneq2dv 5020 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) = 𝑥𝐴 ((1st ↾ (V × V)) ∘ (((1st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
481, 7, 473eqtr4a 2800 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 395   = wceq 1536  wcel 2105  wne 2937  Vcvv 3477  cin 3961  wss 3962  c0 4338  {csn 4630   ciun 4995   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  ccom 5692   Fn wfn 6557  cfv 6562  1st c1st 8010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-1st 8012  df-2nd 8013
This theorem is referenced by:  fpar  8139
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