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Theorem dmmpt2ssx 7280
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 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
dmmpt2ssx 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 dmmpt2ssx
Dummy variables 𝑢 𝑡 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2793 . . . . 5 𝑢𝐵
2 nfcsb1v 3582 . . . . 5 𝑥𝑢 / 𝑥𝐵
3 nfcv 2793 . . . . 5 𝑢𝐶
4 nfcv 2793 . . . . 5 𝑣𝐶
5 nfcsb1v 3582 . . . . 5 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
6 nfcv 2793 . . . . . 6 𝑦𝑢
7 nfcsb1v 3582 . . . . . 6 𝑦𝑣 / 𝑦𝐶
86, 7nfcsb 3584 . . . . 5 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
9 csbeq1a 3575 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
10 csbeq1a 3575 . . . . . 6 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
11 csbeq1a 3575 . . . . . 6 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
1210, 11sylan9eqr 2707 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 6775 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
14 fmpt2x.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 3234 . . . . . . . 8 𝑢 ∈ V
16 vex 3234 . . . . . . . 8 𝑣 ∈ V
1715, 16op1std 7220 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) = 𝑢)
1817csbeq1d 3573 . . . . . 6 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑡) / 𝑦𝐶)
1915, 16op2ndd 7221 . . . . . . . 8 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) = 𝑣)
2019csbeq1d 3573 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
2120csbeq2dv 4025 . . . . . 6 (𝑡 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2218, 21eqtrd 2685 . . . . 5 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2322mpt2mptx 6793 . . . 4 (𝑡 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
2413, 14, 233eqtr4i 2683 . . 3 𝐹 = (𝑡 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶)
2524dmmptss 5669 . 2 dom 𝐹 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
26 nfcv 2793 . . 3 𝑢({𝑥} × 𝐵)
27 nfcv 2793 . . . 4 𝑥{𝑢}
2827, 2nfxp 5176 . . 3 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
29 sneq 4220 . . . 4 (𝑥 = 𝑢 → {𝑥} = {𝑢})
3029, 9xpeq12d 5174 . . 3 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
3126, 28, 30cbviun 4589 . 2 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
3225, 31sseqtr4i 3671 1 dom 𝐹 𝑥𝐴 ({𝑥} × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  csb 3566  wss 3607  {csn 4210  cop 4216   ciun 4552  cmpt 4762   × cxp 5141  dom cdm 5143  cfv 5926  cmpt2 6692  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211
This theorem is referenced by:  mpt2exxg  7289  mpt2xeldm  7382  mpt2xopn0yelv  7384  mpt2xopxnop0  7386  dmcoass  16763  ply1frcl  19731  dvbsss  23711  perfdvf  23712
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