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Theorem dmmpossx2 44781
 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 7749. (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 2955 . . . . 5 𝑢𝐴
2 nfcsb1v 3852 . . . . 5 𝑦𝑢 / 𝑦𝐴
3 nfcv 2955 . . . . 5 𝑢𝐶
4 nfcv 2955 . . . . 5 𝑣𝐶
5 nfcv 2955 . . . . . 6 𝑥𝑢
6 nfcsb1v 3852 . . . . . 6 𝑥𝑣 / 𝑥𝐶
75, 6nfcsbw 3854 . . . . 5 𝑥𝑢 / 𝑦𝑣 / 𝑥𝐶
8 nfcsb1v 3852 . . . . 5 𝑦𝑢 / 𝑦𝑣 / 𝑥𝐶
9 csbeq1a 3842 . . . . 5 (𝑦 = 𝑢𝐴 = 𝑢 / 𝑦𝐴)
10 csbeq1a 3842 . . . . . 6 (𝑥 = 𝑣𝐶 = 𝑣 / 𝑥𝐶)
11 csbeq1a 3842 . . . . . 6 (𝑦 = 𝑢𝑣 / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
1210, 11sylan9eqr 2855 . . . . 5 ((𝑦 = 𝑢𝑥 = 𝑣) → 𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
131, 2, 3, 4, 7, 8, 9, 12cbvmpox2 44780 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
14 dmmpossx2.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 3444 . . . . . . . 8 𝑣 ∈ V
16 vex 3444 . . . . . . . 8 𝑢 ∈ V
1715, 16op2ndd 7685 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) = 𝑢)
1817csbeq1d 3832 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶)
1915, 16op1std 7684 . . . . . . . 8 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) = 𝑣)
2019csbeq1d 3832 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) / 𝑥𝐶 = 𝑣 / 𝑥𝐶)
2120csbeq2dv 3835 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2218, 21eqtrd 2833 . . . . 5 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2322mpomptx2 44779 . . . 4 (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
2413, 14, 233eqtr4i 2831 . . 3 𝐹 = (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶)
2524dmmptss 6063 . 2 dom 𝐹 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
26 nfcv 2955 . . 3 𝑢(𝐴 × {𝑦})
27 nfcv 2955 . . . 4 𝑦{𝑢}
282, 27nfxp 5553 . . 3 𝑦(𝑢 / 𝑦𝐴 × {𝑢})
29 sneq 4535 . . . 4 (𝑦 = 𝑢 → {𝑦} = {𝑢})
309, 29xpeq12d 5551 . . 3 (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (𝑢 / 𝑦𝐴 × {𝑢}))
3126, 28, 30cbviun 4924 . 2 𝑦𝐵 (𝐴 × {𝑦}) = 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
3225, 31sseqtrri 3952 1 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ⦋csb 3828   ⊆ wss 3881  {csn 4525  ⟨cop 4531  ∪ ciun 4882   ↦ cmpt 5111   × cxp 5518  dom cdm 5520  ‘cfv 6325   ∈ cmpo 7138  1st c1st 7672  2nd c2nd 7673 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fv 6333  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675 This theorem is referenced by:  mpoexxg2  44782
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