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Theorem fvproj 8160
Description: Value of a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)
Hypotheses
Ref Expression
fvproj.h 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
fvproj.x (𝜑𝑋𝐴)
fvproj.y (𝜑𝑌𝐵)
Assertion
Ref Expression
fvproj (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem fvproj
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7435 . 2 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2 fvproj.x . . 3 (𝜑𝑋𝐴)
3 fvproj.y . . 3 (𝜑𝑌𝐵)
4 fveq2 6905 . . . . 5 (𝑎 = 𝑋 → (𝐹𝑎) = (𝐹𝑋))
54opeq1d 4878 . . . 4 (𝑎 = 𝑋 → ⟨(𝐹𝑎), (𝐺𝑏)⟩ = ⟨(𝐹𝑋), (𝐺𝑏)⟩)
6 fveq2 6905 . . . . 5 (𝑏 = 𝑌 → (𝐺𝑏) = (𝐺𝑌))
76opeq2d 4879 . . . 4 (𝑏 = 𝑌 → ⟨(𝐹𝑋), (𝐺𝑏)⟩ = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
8 fvproj.h . . . . 5 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
9 fveq2 6905 . . . . . . 7 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
109opeq1d 4878 . . . . . 6 (𝑥 = 𝑎 → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑦)⟩)
11 fveq2 6905 . . . . . . 7 (𝑦 = 𝑏 → (𝐺𝑦) = (𝐺𝑏))
1211opeq2d 4879 . . . . . 6 (𝑦 = 𝑏 → ⟨(𝐹𝑎), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑏)⟩)
1310, 12cbvmpov 7529 . . . . 5 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = (𝑎𝐴, 𝑏𝐵 ↦ ⟨(𝐹𝑎), (𝐺𝑏)⟩)
148, 13eqtri 2764 . . . 4 𝐻 = (𝑎𝐴, 𝑏𝐵 ↦ ⟨(𝐹𝑎), (𝐺𝑏)⟩)
15 opex 5468 . . . 4 ⟨(𝐹𝑋), (𝐺𝑌)⟩ ∈ V
165, 7, 14, 15ovmpo 7594 . . 3 ((𝑋𝐴𝑌𝐵) → (𝑋𝐻𝑌) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
172, 3, 16syl2anc 584 . 2 (𝜑 → (𝑋𝐻𝑌) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
181, 17eqtr3id 2790 1 (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cop 4631  cfv 6560  (class class class)co 7432  cmpo 7434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437
This theorem is referenced by:  fimaproj  8161  ex-fpar  30482  qtophaus  33836
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