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Theorem dmmpossx2 47012
Description: The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpossx 8052. (Contributed by AV, 30-Mar-2019.)
Hypothesis
Ref Expression
dmmpossx2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
dmmpossx2 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵
Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dmmpossx2
Dummy variables 𝑢 𝑡 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2904 . . . . 5 𝑢𝐴
2 nfcsb1v 3919 . . . . 5 𝑦𝑢 / 𝑦𝐴
3 nfcv 2904 . . . . 5 𝑢𝐶
4 nfcv 2904 . . . . 5 𝑣𝐶
5 nfcv 2904 . . . . . 6 𝑥𝑢
6 nfcsb1v 3919 . . . . . 6 𝑥𝑣 / 𝑥𝐶
75, 6nfcsbw 3921 . . . . 5 𝑥𝑢 / 𝑦𝑣 / 𝑥𝐶
8 nfcsb1v 3919 . . . . 5 𝑦𝑢 / 𝑦𝑣 / 𝑥𝐶
9 csbeq1a 3908 . . . . 5 (𝑦 = 𝑢𝐴 = 𝑢 / 𝑦𝐴)
10 csbeq1a 3908 . . . . . 6 (𝑥 = 𝑣𝐶 = 𝑣 / 𝑥𝐶)
11 csbeq1a 3908 . . . . . 6 (𝑦 = 𝑢𝑣 / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
1210, 11sylan9eqr 2795 . . . . 5 ((𝑦 = 𝑢𝑥 = 𝑣) → 𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
131, 2, 3, 4, 7, 8, 9, 12cbvmpox2 47011 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
14 dmmpossx2.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 3479 . . . . . . . 8 𝑣 ∈ V
16 vex 3479 . . . . . . . 8 𝑢 ∈ V
1715, 16op2ndd 7986 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) = 𝑢)
1817csbeq1d 3898 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶)
1915, 16op1std 7985 . . . . . . . 8 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) = 𝑣)
2019csbeq1d 3898 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) / 𝑥𝐶 = 𝑣 / 𝑥𝐶)
2120csbeq2dv 3901 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2218, 21eqtrd 2773 . . . . 5 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2322mpomptx2 47010 . . . 4 (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
2413, 14, 233eqtr4i 2771 . . 3 𝐹 = (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶)
2524dmmptss 6241 . 2 dom 𝐹 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
26 nfcv 2904 . . 3 𝑢(𝐴 × {𝑦})
27 nfcv 2904 . . . 4 𝑦{𝑢}
282, 27nfxp 5710 . . 3 𝑦(𝑢 / 𝑦𝐴 × {𝑢})
29 sneq 4639 . . . 4 (𝑦 = 𝑢 → {𝑦} = {𝑢})
309, 29xpeq12d 5708 . . 3 (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (𝑢 / 𝑦𝐴 × {𝑢}))
3126, 28, 30cbviun 5040 . 2 𝑦𝐵 (𝐴 × {𝑦}) = 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
3225, 31sseqtrri 4020 1 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  csb 3894  wss 3949  {csn 4629  cop 4635   ciun 4998  cmpt 5232   × cxp 5675  dom cdm 5677  cfv 6544  cmpo 7411  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976
This theorem is referenced by:  mpoexxg2  47013
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