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Theorem dmmpossx 8091
Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
fmpox.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
dmmpossx dom 𝐹 𝑥𝐴 ({𝑥} × 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dmmpossx
Dummy variables 𝑢 𝑡 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2905 . . . . 5 𝑢𝐵
2 nfcsb1v 3923 . . . . 5 𝑥𝑢 / 𝑥𝐵
3 nfcv 2905 . . . . 5 𝑢𝐶
4 nfcv 2905 . . . . 5 𝑣𝐶
5 nfcsb1v 3923 . . . . 5 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
6 nfcv 2905 . . . . . 6 𝑦𝑢
7 nfcsb1v 3923 . . . . . 6 𝑦𝑣 / 𝑦𝐶
86, 7nfcsbw 3925 . . . . 5 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
9 csbeq1a 3913 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
10 csbeq1a 3913 . . . . . 6 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
11 csbeq1a 3913 . . . . . 6 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
1210, 11sylan9eqr 2799 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
131, 2, 3, 4, 5, 8, 9, 12cbvmpox 7526 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
14 fmpox.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 3484 . . . . . . . 8 𝑢 ∈ V
16 vex 3484 . . . . . . . 8 𝑣 ∈ V
1715, 16op1std 8024 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) = 𝑢)
1817csbeq1d 3903 . . . . . 6 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑡) / 𝑦𝐶)
1915, 16op2ndd 8025 . . . . . . . 8 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) = 𝑣)
2019csbeq1d 3903 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
2120csbeq2dv 3906 . . . . . 6 (𝑡 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2218, 21eqtrd 2777 . . . . 5 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2322mpomptx 7546 . . . 4 (𝑡 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
2413, 14, 233eqtr4i 2775 . . 3 𝐹 = (𝑡 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶)
2524dmmptss 6261 . 2 dom 𝐹 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
26 nfcv 2905 . . 3 𝑢({𝑥} × 𝐵)
27 nfcv 2905 . . . 4 𝑥{𝑢}
2827, 2nfxp 5718 . . 3 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
29 sneq 4636 . . . 4 (𝑥 = 𝑢 → {𝑥} = {𝑢})
3029, 9xpeq12d 5716 . . 3 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
3126, 28, 30cbviun 5036 . 2 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
3225, 31sseqtrri 4033 1 dom 𝐹 𝑥𝐴 ({𝑥} × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  csb 3899  wss 3951  {csn 4626  cop 4632   ciun 4991  cmpt 5225   × cxp 5683  dom cdm 5685  cfv 6561  cmpo 7433  1st c1st 8012  2nd c2nd 8013
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015
This theorem is referenced by:  mpoexxg  8100  mpoxeldm  8236  mpoxopn0yelv  8238  mpoxopxnop0  8240  dmcoass  18111  ply1frcl  22322  dvbsss  25937  perfdvf  25938
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