MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmmpossx Structured version   Visualization version   GIF version

Theorem dmmpossx 7999
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 2908 . . . . 5 𝑢𝐵
2 nfcsb1v 3881 . . . . 5 𝑥𝑢 / 𝑥𝐵
3 nfcv 2908 . . . . 5 𝑢𝐶
4 nfcv 2908 . . . . 5 𝑣𝐶
5 nfcsb1v 3881 . . . . 5 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
6 nfcv 2908 . . . . . 6 𝑦𝑢
7 nfcsb1v 3881 . . . . . 6 𝑦𝑣 / 𝑦𝐶
86, 7nfcsbw 3883 . . . . 5 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
9 csbeq1a 3870 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
10 csbeq1a 3870 . . . . . 6 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
11 csbeq1a 3870 . . . . . 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 7451 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
14 fmpox.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 3450 . . . . . . . 8 𝑢 ∈ V
16 vex 3450 . . . . . . . 8 𝑣 ∈ V
1715, 16op1std 7932 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) = 𝑢)
1817csbeq1d 3860 . . . . . 6 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑡) / 𝑦𝐶)
1915, 16op2ndd 7933 . . . . . . . 8 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) = 𝑣)
2019csbeq1d 3860 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
2120csbeq2dv 3863 . . . . . 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 7470 . . . 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 6194 . 2 dom 𝐹 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
26 nfcv 2908 . . 3 𝑢({𝑥} × 𝐵)
27 nfcv 2908 . . . 4 𝑥{𝑢}
2827, 2nfxp 5667 . . 3 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
29 sneq 4597 . . . 4 (𝑥 = 𝑢 → {𝑥} = {𝑢})
3029, 9xpeq12d 5665 . . 3 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
3126, 28, 30cbviun 4997 . 2 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
3225, 31sseqtrri 3982 1 dom 𝐹 𝑥𝐴 ({𝑥} × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  csb 3856  wss 3911  {csn 4587  cop 4593   ciun 4955  cmpt 5189   × cxp 5632  dom cdm 5634  cfv 6497  cmpo 7360  1st c1st 7920  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923
This theorem is referenced by:  mpoexxg  8009  mpoxeldm  8143  mpoxopn0yelv  8145  mpoxopxnop0  8147  dmcoass  17953  ply1frcl  21687  dvbsss  25269  perfdvf  25270
  Copyright terms: Public domain W3C validator