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Theorem dmmpt2ssx 7515
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 2934 . . . . 5 𝑢𝐵
2 nfcsb1v 3767 . . . . 5 𝑥𝑢 / 𝑥𝐵
3 nfcv 2934 . . . . 5 𝑢𝐶
4 nfcv 2934 . . . . 5 𝑣𝐶
5 nfcsb1v 3767 . . . . 5 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
6 nfcv 2934 . . . . . 6 𝑦𝑢
7 nfcsb1v 3767 . . . . . 6 𝑦𝑣 / 𝑦𝐶
86, 7nfcsb 3769 . . . . 5 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
9 csbeq1a 3760 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
10 csbeq1a 3760 . . . . . 6 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
11 csbeq1a 3760 . . . . . 6 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
1210, 11sylan9eqr 2836 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 7010 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
14 fmpt2x.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 3401 . . . . . . . 8 𝑢 ∈ V
16 vex 3401 . . . . . . . 8 𝑣 ∈ V
1715, 16op1std 7455 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) = 𝑢)
1817csbeq1d 3758 . . . . . 6 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑡) / 𝑦𝐶)
1915, 16op2ndd 7456 . . . . . . . 8 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) = 𝑣)
2019csbeq1d 3758 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd𝑡) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
2120csbeq2dv 4217 . . . . . 6 (𝑡 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2218, 21eqtrd 2814 . . . . 5 (𝑡 = ⟨𝑢, 𝑣⟩ → (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2322mpt2mptx 7028 . . . 4 (𝑡 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
2413, 14, 233eqtr4i 2812 . . 3 𝐹 = (𝑡 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑡) / 𝑥(2nd𝑡) / 𝑦𝐶)
2524dmmptss 5885 . 2 dom 𝐹 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
26 nfcv 2934 . . 3 𝑢({𝑥} × 𝐵)
27 nfcv 2934 . . . 4 𝑥{𝑢}
2827, 2nfxp 5388 . . 3 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
29 sneq 4408 . . . 4 (𝑥 = 𝑢 → {𝑥} = {𝑢})
3029, 9xpeq12d 5386 . . 3 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
3126, 28, 30cbviun 4790 . 2 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
3225, 31sseqtr4i 3857 1 dom 𝐹 𝑥𝐴 ({𝑥} × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  csb 3751  wss 3792  {csn 4398  cop 4404   ciun 4753  cmpt 4965   × cxp 5353  dom cdm 5355  cfv 6135  cmpt2 6924  1st c1st 7443  2nd c2nd 7444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fv 6143  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446
This theorem is referenced by:  mpt2exxg  7524  mpt2xeldm  7619  mpt2xopn0yelv  7621  mpt2xopxnop0  7623  dmcoass  17101  ply1frcl  20079  dvbsss  24103  perfdvf  24104
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