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Theorem ucnimalem 24203
Description: Reformulate the 𝐺 function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Hypotheses
Ref Expression
ucnprima.1 (πœ‘ β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
ucnprima.2 (πœ‘ β†’ 𝑉 ∈ (UnifOnβ€˜π‘Œ))
ucnprima.3 (πœ‘ β†’ 𝐹 ∈ (π‘ˆ Cnu𝑉))
ucnprima.4 (πœ‘ β†’ π‘Š ∈ 𝑉)
ucnprima.5 𝐺 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(πΉβ€˜π‘₯), (πΉβ€˜π‘¦)⟩)
Assertion
Ref Expression
ucnimalem 𝐺 = (𝑝 ∈ (𝑋 Γ— 𝑋) ↦ ⟨(πΉβ€˜(1st β€˜π‘)), (πΉβ€˜(2nd β€˜π‘))⟩)
Distinct variable groups:   π‘₯,𝑝,𝑦,𝐹   𝑋,𝑝,π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑝)   π‘ˆ(π‘₯,𝑦,𝑝)   𝐺(π‘₯,𝑦,𝑝)   𝑉(π‘₯,𝑦,𝑝)   π‘Š(π‘₯,𝑦,𝑝)   π‘Œ(π‘₯,𝑦,𝑝)

Proof of Theorem ucnimalem
StepHypRef Expression
1 ucnprima.5 . 2 𝐺 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(πΉβ€˜π‘₯), (πΉβ€˜π‘¦)⟩)
2 vex 3467 . . . . . 6 π‘₯ ∈ V
3 vex 3467 . . . . . 6 𝑦 ∈ V
42, 3op1std 8001 . . . . 5 (𝑝 = ⟨π‘₯, π‘¦βŸ© β†’ (1st β€˜π‘) = π‘₯)
54fveq2d 6896 . . . 4 (𝑝 = ⟨π‘₯, π‘¦βŸ© β†’ (πΉβ€˜(1st β€˜π‘)) = (πΉβ€˜π‘₯))
62, 3op2ndd 8002 . . . . 5 (𝑝 = ⟨π‘₯, π‘¦βŸ© β†’ (2nd β€˜π‘) = 𝑦)
76fveq2d 6896 . . . 4 (𝑝 = ⟨π‘₯, π‘¦βŸ© β†’ (πΉβ€˜(2nd β€˜π‘)) = (πΉβ€˜π‘¦))
85, 7opeq12d 4877 . . 3 (𝑝 = ⟨π‘₯, π‘¦βŸ© β†’ ⟨(πΉβ€˜(1st β€˜π‘)), (πΉβ€˜(2nd β€˜π‘))⟩ = ⟨(πΉβ€˜π‘₯), (πΉβ€˜π‘¦)⟩)
98mpompt 7531 . 2 (𝑝 ∈ (𝑋 Γ— 𝑋) ↦ ⟨(πΉβ€˜(1st β€˜π‘)), (πΉβ€˜(2nd β€˜π‘))⟩) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(πΉβ€˜π‘₯), (πΉβ€˜π‘¦)⟩)
101, 9eqtr4i 2756 1 𝐺 = (𝑝 ∈ (𝑋 Γ— 𝑋) ↦ ⟨(πΉβ€˜(1st β€˜π‘)), (πΉβ€˜(2nd β€˜π‘))⟩)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4630   ↦ cmpt 5226   Γ— cxp 5670  β€˜cfv 6543  (class class class)co 7416   ∈ cmpo 7418  1st c1st 7989  2nd c2nd 7990  UnifOncust 24122   Cnucucn 24198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992
This theorem is referenced by:  ucnima  24204
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