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Theorem ucnimalem 22881
Description: Reformulate the 𝐺 function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Hypotheses
Ref Expression
ucnprima.1 (𝜑𝑈 ∈ (UnifOn‘𝑋))
ucnprima.2 (𝜑𝑉 ∈ (UnifOn‘𝑌))
ucnprima.3 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
ucnprima.4 (𝜑𝑊𝑉)
ucnprima.5 𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
Assertion
Ref Expression
ucnimalem 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
Distinct variable groups:   𝑥,𝑝,𝑦,𝐹   𝑋,𝑝,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑝)   𝑈(𝑥,𝑦,𝑝)   𝐺(𝑥,𝑦,𝑝)   𝑉(𝑥,𝑦,𝑝)   𝑊(𝑥,𝑦,𝑝)   𝑌(𝑥,𝑦,𝑝)

Proof of Theorem ucnimalem
StepHypRef Expression
1 ucnprima.5 . 2 𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
2 vex 3496 . . . . . 6 𝑥 ∈ V
3 vex 3496 . . . . . 6 𝑦 ∈ V
42, 3op1std 7691 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → (1st𝑝) = 𝑥)
54fveq2d 6667 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑝)) = (𝐹𝑥))
62, 3op2ndd 7692 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → (2nd𝑝) = 𝑦)
76fveq2d 6667 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd𝑝)) = (𝐹𝑦))
85, 7opeq12d 4803 . . 3 (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
98mpompt 7258 . 2 (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩) = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
101, 9eqtr4i 2845 1 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cop 4565  cmpt 5137   × cxp 5546  cfv 6348  (class class class)co 7148  cmpo 7150  1st c1st 7679  2nd c2nd 7680  UnifOncust 22800   Cnucucn 22876
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682
This theorem is referenced by:  ucnima  22882
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