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Theorem dmmpt2ssx 5767
Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
fmpt2x.1 𝐹 = (x A, y B𝐶)
Assertion
Ref Expression
dmmpt2ssx dom 𝐹 x A ({x} × B)
Distinct variable groups:   x,y,A   y,B
Allowed substitution hints:   B(x)   𝐶(x,y)   𝐹(x,y)

Proof of Theorem dmmpt2ssx
Dummy variables u 𝑡 v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2175 . . . . 5 uB
2 nfcsb1v 2876 . . . . 5 xu / xB
3 nfcv 2175 . . . . 5 u𝐶
4 nfcv 2175 . . . . 5 v𝐶
5 nfcsb1v 2876 . . . . 5 xu / xv / y𝐶
6 nfcv 2175 . . . . . 6 yu
7 nfcsb1v 2876 . . . . . 6 yv / y𝐶
86, 7nfcsb 2878 . . . . 5 yu / xv / y𝐶
9 csbeq1a 2854 . . . . 5 (x = uB = u / xB)
10 csbeq1a 2854 . . . . . 6 (y = v𝐶 = v / y𝐶)
11 csbeq1a 2854 . . . . . 6 (x = uv / y𝐶 = u / xv / y𝐶)
1210, 11sylan9eqr 2091 . . . . 5 ((x = u y = v) → 𝐶 = u / xv / y𝐶)
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 5524 . . . 4 (x A, y B𝐶) = (u A, v u / xBu / xv / y𝐶)
14 fmpt2x.1 . . . 4 𝐹 = (x A, y B𝐶)
15 vex 2554 . . . . . . . 8 u V
16 vex 2554 . . . . . . . 8 v V
1715, 16op1std 5717 . . . . . . 7 (𝑡 = ⟨u, v⟩ → (1st𝑡) = u)
1817csbeq1d 2852 . . . . . 6 (𝑡 = ⟨u, v⟩ → (1st𝑡) / x(2nd𝑡) / y𝐶 = u / x(2nd𝑡) / y𝐶)
1915, 16op2ndd 5718 . . . . . . . 8 (𝑡 = ⟨u, v⟩ → (2nd𝑡) = v)
2019csbeq1d 2852 . . . . . . 7 (𝑡 = ⟨u, v⟩ → (2nd𝑡) / y𝐶 = v / y𝐶)
2120csbeq2dv 2869 . . . . . 6 (𝑡 = ⟨u, v⟩ → u / x(2nd𝑡) / y𝐶 = u / xv / y𝐶)
2218, 21eqtrd 2069 . . . . 5 (𝑡 = ⟨u, v⟩ → (1st𝑡) / x(2nd𝑡) / y𝐶 = u / xv / y𝐶)
2322mpt2mptx 5537 . . . 4 (𝑡 u A ({u} × u / xB) ↦ (1st𝑡) / x(2nd𝑡) / y𝐶) = (u A, v u / xBu / xv / y𝐶)
2413, 14, 233eqtr4i 2067 . . 3 𝐹 = (𝑡 u A ({u} × u / xB) ↦ (1st𝑡) / x(2nd𝑡) / y𝐶)
2524dmmptss 4760 . 2 dom 𝐹 u A ({u} × u / xB)
26 nfcv 2175 . . 3 u({x} × B)
27 nfcv 2175 . . . 4 x{u}
2827, 2nfxp 4314 . . 3 x({u} × u / xB)
29 sneq 3378 . . . 4 (x = u → {x} = {u})
3029, 9xpeq12d 4313 . . 3 (x = u → ({x} × B) = ({u} × u / xB))
3126, 28, 30cbviun 3685 . 2 x A ({x} × B) = u A ({u} × u / xB)
3225, 31sseqtr4i 2972 1 dom 𝐹 x A ({x} × B)
Colors of variables: wff set class
Syntax hints:   = wceq 1242  csb 2846  wss 2911  {csn 3367  cop 3370   ciun 3648  cmpt 3809   × cxp 4286  dom cdm 4288  cfv 4845  cmpt2 5457  1st c1st 5707  2nd c2nd 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  mpt2exxg  5775  mpt2xopn0yelv  5795
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