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Theorem ucnimalem 22065
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 3198 . . . . . 6 𝑥 ∈ V
3 vex 3198 . . . . . 6 𝑦 ∈ V
42, 3op1std 7163 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → (1st𝑝) = 𝑥)
54fveq2d 6182 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑝)) = (𝐹𝑥))
62, 3op2ndd 7164 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → (2nd𝑝) = 𝑦)
76fveq2d 6182 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd𝑝)) = (𝐹𝑦))
85, 7opeq12d 4401 . . 3 (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
98mpt2mpt 6737 . 2 (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩) = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
101, 9eqtr4i 2645 1 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cop 4174  cmpt 4720   × cxp 5102  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  UnifOncust 21984   Cnucucn 22060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154
This theorem is referenced by:  ucnima  22066
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