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Theorem dfmpo 7772
 Description: Alternate definition for the maps-to notation df-mpo 7135 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfmpo.1 𝐶 ∈ V
Assertion
Ref Expression
dfmpo (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem dfmpo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 mpompts 7738 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶)
2 dfmpo.1 . . . . 5 𝐶 ∈ V
32csbex 5188 . . . 4 (2nd𝑤) / 𝑦𝐶 ∈ V
43csbex 5188 . . 3 (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 ∈ V
54dfmpt 6879 . 2 (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶) = 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
6 nfcv 2974 . . . . 5 𝑥𝑤
7 nfcsb1v 3881 . . . . 5 𝑥(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
86, 7nfop 4792 . . . 4 𝑥𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
98nfsn 4616 . . 3 𝑥{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
10 nfcv 2974 . . . . 5 𝑦𝑤
11 nfcv 2974 . . . . . 6 𝑦(1st𝑤)
12 nfcsb1v 3881 . . . . . 6 𝑦(2nd𝑤) / 𝑦𝐶
1311, 12nfcsbw 3883 . . . . 5 𝑦(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1410, 13nfop 4792 . . . 4 𝑦𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1514nfsn 4616 . . 3 𝑦{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
16 nfcv 2974 . . 3 𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
17 id 22 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
18 csbopeq1a 7724 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 = 𝐶)
1917, 18opeq12d 4784 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
2019sneqd 4552 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
219, 15, 16, 20iunxpf 5692 . 2 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
221, 5, 213eqtri 2848 1 (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2115  Vcvv 3471  ⦋csb 3857  {csn 4540  ⟨cop 4546  ∪ ciun 4892   ↦ cmpt 5119   × cxp 5526  ‘cfv 6328   ∈ cmpo 7132  1st c1st 7662  2nd c2nd 7663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665 This theorem is referenced by:  fpar  7786
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