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Theorem fparlem4 7515
Description: Lemma for fpar 7516. (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 5862 . 2 ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
2 inss1 4026 . . . . 5 (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ dom 𝐺
3 fndm 6199 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3syl5sseq 3847 . . . 4 (𝐺 Fn 𝐵 → (dom 𝐺 ∩ ran (2nd ↾ (V × V))) ⊆ 𝐵)
5 dfco2a 5852 . . . 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 5486 . 2 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ 𝑦𝐵 (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
8 inss1 4026 . . . . . . . . 9 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ dom ({(𝐺𝑦)} × (V × {𝑦}))
9 dmxpss 5780 . . . . . . . . 9 dom ({(𝐺𝑦)} × (V × {𝑦})) ⊆ {(𝐺𝑦)}
108, 9sstri 3805 . . . . . . . 8 (dom ({(𝐺𝑦)} × (V × {𝑦})) ∩ ran (2nd ↾ (V × V))) ⊆ {(𝐺𝑦)}
11 dfco2a 5852 . . . . . . . 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 6422 . . . . . . . 8 (𝐺𝑦) ∈ V
14 fparlem2 7513 . . . . . . . . . 10 ((2nd ↾ (V × V)) “ {𝑥}) = (V × {𝑥})
15 sneq 4376 . . . . . . . . . . 11 (𝑥 = (𝐺𝑦) → {𝑥} = {(𝐺𝑦)})
1615xpeq2d 5340 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (V × {𝑥}) = (V × {(𝐺𝑦)}))
1714, 16syl5eq 2843 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → ((2nd ↾ (V × V)) “ {𝑥}) = (V × {(𝐺𝑦)}))
1815imaeq2d 5681 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}))
19 df-ima 5323 . . . . . . . . . . 11 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)})
20 ssid 3817 . . . . . . . . . . . . . 14 {(𝐺𝑦)} ⊆ {(𝐺𝑦)}
21 xpssres 5641 . . . . . . . . . . . . . 14 ({(𝐺𝑦)} ⊆ {(𝐺𝑦)} → (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦})))
2220, 21ax-mp 5 . . . . . . . . . . . . 13 (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ({(𝐺𝑦)} × (V × {𝑦}))
2322rneqi 5553 . . . . . . . . . . . 12 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = ran ({(𝐺𝑦)} × (V × {𝑦}))
2413snnz 4495 . . . . . . . . . . . . 13 {(𝐺𝑦)} ≠ ∅
25 rnxp 5779 . . . . . . . . . . . . 13 ({(𝐺𝑦)} ≠ ∅ → ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦}))
2624, 25ax-mp 5 . . . . . . . . . . . 12 ran ({(𝐺𝑦)} × (V × {𝑦})) = (V × {𝑦})
2723, 26eqtri 2819 . . . . . . . . . . 11 ran (({(𝐺𝑦)} × (V × {𝑦})) ↾ {(𝐺𝑦)}) = (V × {𝑦})
2819, 27eqtri 2819 . . . . . . . . . 10 (({(𝐺𝑦)} × (V × {𝑦})) “ {(𝐺𝑦)}) = (V × {𝑦})
2918, 28syl6eq 2847 . . . . . . . . 9 (𝑥 = (𝐺𝑦) → (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥}) = (V × {𝑦}))
3017, 29xpeq12d 5341 . . . . . . . 8 (𝑥 = (𝐺𝑦) → (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦})))
3113, 30iunxsn 4791 . . . . . . 7 𝑥 ∈ {(𝐺𝑦)} (((2nd ↾ (V × V)) “ {𝑥}) × (({(𝐺𝑦)} × (V × {𝑦})) “ {𝑥})) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3212, 31eqtri 2819 . . . . . 6 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
3332cnveqi 5498 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((V × {(𝐺𝑦)}) × (V × {𝑦}))
34 cnvco 5509 . . . . 5 (({(𝐺𝑦)} × (V × {𝑦})) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦})))
35 cnvxp 5766 . . . . 5 ((V × {(𝐺𝑦)}) × (V × {𝑦})) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
3633, 34, 353eqtr3i 2827 . . . 4 ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((V × {𝑦}) × (V × {(𝐺𝑦)}))
37 fparlem2 7513 . . . . . . . . 9 ((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})
3837xpeq2i 5337 . . . . . . . 8 ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ({(𝐺𝑦)} × (V × {𝑦}))
39 fnsnfv 6481 . . . . . . . . 9 ((𝐺 Fn 𝐵𝑦𝐵) → {(𝐺𝑦)} = (𝐺 “ {𝑦}))
4039xpeq1d 5339 . . . . . . . 8 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × ((2nd ↾ (V × V)) “ {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4138, 40syl5eqr 2845 . . . . . . 7 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
4241cnveqd 5499 . . . . . 6 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})))
43 cnvxp 5766 . . . . . 6 ((𝐺 “ {𝑦}) × ((2nd ↾ (V × V)) “ {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))
4442, 43syl6eq 2847 . . . . 5 ((𝐺 Fn 𝐵𝑦𝐵) → ({(𝐺𝑦)} × (V × {𝑦})) = (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦})))
4544coeq2d 5486 . . . 4 ((𝐺 Fn 𝐵𝑦𝐵) → ((2nd ↾ (V × V)) ∘ ({(𝐺𝑦)} × (V × {𝑦}))) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4636, 45syl5eqr 2845 . . 3 ((𝐺 Fn 𝐵𝑦𝐵) → ((V × {𝑦}) × (V × {(𝐺𝑦)})) = ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
4746iuneq2dv 4730 . 2 (𝐺 Fn 𝐵 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})) = 𝑦𝐵 ((2nd ↾ (V × V)) ∘ (((2nd ↾ (V × V)) “ {𝑦}) × (𝐺 “ {𝑦}))))
481, 7, 473eqtr4a 2857 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 385   = wceq 1653  wcel 2157  wne 2969  Vcvv 3383  cin 3766  wss 3767  c0 4113  {csn 4366   ciun 4708   × cxp 5308  ccnv 5309  dom cdm 5310  ran crn 5311  cres 5312  cima 5313  ccom 5314   Fn wfn 6094  cfv 6099  2nd c2nd 7398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-1st 7399  df-2nd 7400
This theorem is referenced by:  fpar  7516
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