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Theorem dmmpt2ssx2 41400
Description: The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 7180. (Contributed by AV, 30-Mar-2019.)
Hypothesis
Ref Expression
dmmpt2ssx2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
dmmpt2ssx2 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵
Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dmmpt2ssx2
Dummy variables 𝑢 𝑡 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2761 . . . . 5 𝑢𝐴
2 nfcsb1v 3530 . . . . 5 𝑦𝑢 / 𝑦𝐴
3 nfcv 2761 . . . . 5 𝑢𝐶
4 nfcv 2761 . . . . 5 𝑣𝐶
5 nfcv 2761 . . . . . 6 𝑥𝑢
6 nfcsb1v 3530 . . . . . 6 𝑥𝑣 / 𝑥𝐶
75, 6nfcsb 3532 . . . . 5 𝑥𝑢 / 𝑦𝑣 / 𝑥𝐶
8 nfcsb1v 3530 . . . . 5 𝑦𝑢 / 𝑦𝑣 / 𝑥𝐶
9 csbeq1a 3523 . . . . 5 (𝑦 = 𝑢𝐴 = 𝑢 / 𝑦𝐴)
10 csbeq1a 3523 . . . . . 6 (𝑥 = 𝑣𝐶 = 𝑣 / 𝑥𝐶)
11 csbeq1a 3523 . . . . . 6 (𝑦 = 𝑢𝑣 / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
1210, 11sylan9eqr 2677 . . . . 5 ((𝑦 = 𝑢𝑥 = 𝑣) → 𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
131, 2, 3, 4, 7, 8, 9, 12cbvmpt2x2 41399 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
14 dmmpt2ssx2.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 3189 . . . . . . . 8 𝑣 ∈ V
16 vex 3189 . . . . . . . 8 𝑢 ∈ V
1715, 16op2ndd 7124 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) = 𝑢)
1817csbeq1d 3521 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶)
1915, 16op1std 7123 . . . . . . . 8 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) = 𝑣)
2019csbeq1d 3521 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) / 𝑥𝐶 = 𝑣 / 𝑥𝐶)
2120csbeq2dv 3964 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2218, 21eqtrd 2655 . . . . 5 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2322mpt2mptx2 41398 . . . 4 (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
2413, 14, 233eqtr4i 2653 . . 3 𝐹 = (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶)
2524dmmptss 5590 . 2 dom 𝐹 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
26 nfcv 2761 . . 3 𝑢(𝐴 × {𝑦})
27 nfcv 2761 . . . 4 𝑦{𝑢}
282, 27nfxp 5102 . . 3 𝑦(𝑢 / 𝑦𝐴 × {𝑢})
29 sneq 4158 . . . 4 (𝑦 = 𝑢 → {𝑦} = {𝑢})
309, 29xpeq12d 5100 . . 3 (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (𝑢 / 𝑦𝐴 × {𝑢}))
3126, 28, 30cbviun 4523 . 2 𝑦𝐵 (𝐴 × {𝑦}) = 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
3225, 31sseqtr4i 3617 1 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  csb 3514  wss 3555  {csn 4148  cop 4154   ciun 4485  cmpt 4673   × cxp 5072  dom cdm 5074  cfv 5847  cmpt2 6606  1st c1st 7111  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114
This theorem is referenced by:  mpt2exxg2  41401
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