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Theorem dmmpt2ssx2 41908
 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 7124. (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 2751 . . . . 5 𝑢𝐴
2 nfcsb1v 3515 . . . . 5 𝑦𝑢 / 𝑦𝐴
3 nfcv 2751 . . . . 5 𝑢𝐶
4 nfcv 2751 . . . . 5 𝑣𝐶
5 nfcv 2751 . . . . . 6 𝑥𝑢
6 nfcsb1v 3515 . . . . . 6 𝑥𝑣 / 𝑥𝐶
75, 6nfcsb 3517 . . . . 5 𝑥𝑢 / 𝑦𝑣 / 𝑥𝐶
8 nfcsb1v 3515 . . . . 5 𝑦𝑢 / 𝑦𝑣 / 𝑥𝐶
9 csbeq1a 3508 . . . . 5 (𝑦 = 𝑢𝐴 = 𝑢 / 𝑦𝐴)
10 csbeq1a 3508 . . . . . 6 (𝑥 = 𝑣𝐶 = 𝑣 / 𝑥𝐶)
11 csbeq1a 3508 . . . . . 6 (𝑦 = 𝑢𝑣 / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
1210, 11sylan9eqr 2666 . . . . 5 ((𝑦 = 𝑢𝑥 = 𝑣) → 𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
131, 2, 3, 4, 7, 8, 9, 12cbvmpt2x2 41907 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
14 dmmpt2ssx2.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
15 vex 3176 . . . . . . . 8 𝑣 ∈ V
16 vex 3176 . . . . . . . 8 𝑢 ∈ V
1715, 16op2ndd 7070 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) = 𝑢)
1817csbeq1d 3506 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶)
1915, 16op1std 7069 . . . . . . . 8 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) = 𝑣)
2019csbeq1d 3506 . . . . . . 7 (𝑡 = ⟨𝑣, 𝑢⟩ → (1st𝑡) / 𝑥𝐶 = 𝑣 / 𝑥𝐶)
2120csbeq2dv 3944 . . . . . 6 (𝑡 = ⟨𝑣, 𝑢⟩ → 𝑢 / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2218, 21eqtrd 2644 . . . . 5 (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶 = 𝑢 / 𝑦𝑣 / 𝑥𝐶)
2322mpt2mptx2 41906 . . . 4 (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶) = (𝑣𝑢 / 𝑦𝐴, 𝑢𝐵𝑢 / 𝑦𝑣 / 𝑥𝐶)
2413, 14, 233eqtr4i 2642 . . 3 𝐹 = (𝑡 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢}) ↦ (2nd𝑡) / 𝑦(1st𝑡) / 𝑥𝐶)
2524dmmptss 5548 . 2 dom 𝐹 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
26 nfcv 2751 . . 3 𝑢(𝐴 × {𝑦})
27 nfcv 2751 . . . 4 𝑦{𝑢}
282, 27nfxp 5066 . . 3 𝑦(𝑢 / 𝑦𝐴 × {𝑢})
29 sneq 4135 . . . 4 (𝑦 = 𝑢 → {𝑦} = {𝑢})
309, 29xpeq12d 5064 . . 3 (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (𝑢 / 𝑦𝐴 × {𝑢}))
3126, 28, 30cbviun 4493 . 2 𝑦𝐵 (𝐴 × {𝑦}) = 𝑢𝐵 (𝑢 / 𝑦𝐴 × {𝑢})
3225, 31sseqtr4i 3601 1 dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ⦋csb 3499   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ‘cfv 5804   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060 This theorem is referenced by:  mpt2exxg2  41909
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