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Theorem dmmpossx2 47100
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 8054. (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 2901 . . . . 5 𝑢𝐴
2 nfcsb1v 3917 . . . . 5 𝑦𝑢 / 𝑦𝐴
3 nfcv 2901 . . . . 5 𝑢𝐶
4 nfcv 2901 . . . . 5 𝑣𝐶
5 nfcv 2901 . . . . . 6 𝑥𝑢
6 nfcsb1v 3917 . . . . . 6 𝑥𝑣 / 𝑥𝐶
75, 6nfcsbw 3919 . . . . 5 𝑥𝑢 / 𝑦𝑣 / 𝑥𝐶
8 nfcsb1v 3917 . . . . 5 𝑦𝑢 / 𝑦𝑣 / 𝑥𝐶
9 csbeq1a 3906 . . . . 5 (𝑦 = 𝑢𝐴 = 𝑢 / 𝑦𝐴)
10 csbeq1a 3906 . . . . . 6 (𝑥 = 𝑣𝐶 = 𝑣 / 𝑥𝐶)
11 csbeq1a 3906 . . . . . 6 (𝑦 = 𝑢𝑣 / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
1210, 11sylan9eqr 2792 . . . . 5 ((𝑦 = 𝑢𝑥 = 𝑣) → 𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
131, 2, 3, 4, 7, 8, 9, 12cbvmpox2 47099 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
14 dmmpossx2.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 3476 . . . . . . . 8 𝑣 ∈ V
16 vex 3476 . . . . . . . 8 𝑢 ∈ V
1715, 16op2ndd 7988 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) = 𝑢)
1817csbeq1d 3896 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶)
1915, 16op1std 7987 . . . . . . . 8 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) = 𝑣)
2019csbeq1d 3896 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) / 𝑥𝐶 = 𝑣 / 𝑥𝐶)
2120csbeq2dv 3899 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2218, 21eqtrd 2770 . . . . 5 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2322mpomptx2 47098 . . . 4 (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
2413, 14, 233eqtr4i 2768 . . 3 𝐹 = (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶)
2524dmmptss 6239 . 2 dom 𝐹 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
26 nfcv 2901 . . 3 𝑢(𝐴 × {𝑦})
27 nfcv 2901 . . . 4 𝑦{𝑢}
282, 27nfxp 5708 . . 3 𝑦(𝑢 / 𝑦𝐴 × {𝑢})
29 sneq 4637 . . . 4 (𝑦 = 𝑢 → {𝑦} = {𝑢})
309, 29xpeq12d 5706 . . 3 (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (𝑢 / 𝑦𝐴 × {𝑢}))
3126, 28, 30cbviun 5038 . 2 𝑦𝐵 (𝐴 × {𝑦}) = 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
3225, 31sseqtrri 4018 1 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  csb 3892  wss 3947  {csn 4627  cop 4633   ciun 4996  cmpt 5230   × cxp 5673  dom cdm 5675  cfv 6542  cmpo 7413  1st c1st 7975  2nd c2nd 7976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fv 6550  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978
This theorem is referenced by:  mpoexxg2  47101
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