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Theorem dmmpossx 6106
 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 2282 . . . . 5 𝑢𝐵
2 nfcsb1v 3041 . . . . 5 𝑥𝑢 / 𝑥𝐵
3 nfcv 2282 . . . . 5 𝑢𝐶
4 nfcv 2282 . . . . 5 𝑣𝐶
5 nfcsb1v 3041 . . . . 5 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
6 nfcv 2282 . . . . . 6 𝑦𝑢
7 nfcsb1v 3041 . . . . . 6 𝑦𝑣 / 𝑦𝐶
86, 7nfcsb 3043 . . . . 5 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
9 csbeq1a 3017 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
10 csbeq1a 3017 . . . . . 6 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
11 csbeq1a 3017 . . . . . 6 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
1210, 11sylan9eqr 2195 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
131, 2, 3, 4, 5, 8, 9, 12cbvmpox 5858 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
14 fmpox.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 2693 . . . . . . . 8 𝑢 ∈ V
16 vex 2693 . . . . . . . 8 𝑣 ∈ V
1715, 16op1std 6055 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) = 𝑢)
1817csbeq1d 3015 . . . . . 6 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑡) / 𝑦𝐶)
1915, 16op2ndd 6056 . . . . . . . 8 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) = 𝑣)
2019csbeq1d 3015 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
2120csbeq2dv 3034 . . . . . 6 (𝑡 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2218, 21eqtrd 2173 . . . . 5 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2322mpomptx 5871 . . . 4 (𝑡 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
2413, 14, 233eqtr4i 2171 . . 3 𝐹 = (𝑡 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶)
2524dmmptss 5044 . 2 dom 𝐹 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
26 nfcv 2282 . . 3 𝑢({𝑥} × 𝐵)
27 nfcv 2282 . . . 4 𝑥{𝑢}
2827, 2nfxp 4575 . . 3 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
29 sneq 3544 . . . 4 (𝑥 = 𝑢 → {𝑥} = {𝑢})
3029, 9xpeq12d 4573 . . 3 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
3126, 28, 30cbviun 3859 . 2 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
3225, 31sseqtrri 3138 1 dom 𝐹 𝑥𝐴 ({𝑥} × 𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1332  ⦋csb 3008   ⊆ wss 3077  {csn 3533  ⟨cop 3536  ∪ ciun 3822   ↦ cmpt 3998   × cxp 4546  dom cdm 4548  ‘cfv 5132   ∈ cmpo 5785  1st c1st 6045  2nd c2nd 6046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fv 5140  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048 This theorem is referenced by:  mpoexxg  6117  mpoxopn0yelv  6145
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