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Theorem fvproj 7847
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 7167 . 2 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2 fvproj.x . . 3 (𝜑𝑋𝐴)
3 fvproj.y . . 3 (𝜑𝑌𝐵)
4 fveq2 6668 . . . . 5 (𝑎 = 𝑋 → (𝐹𝑎) = (𝐹𝑋))
54opeq1d 4764 . . . 4 (𝑎 = 𝑋 → ⟨(𝐹𝑎), (𝐺𝑏)⟩ = ⟨(𝐹𝑋), (𝐺𝑏)⟩)
6 fveq2 6668 . . . . 5 (𝑏 = 𝑌 → (𝐺𝑏) = (𝐺𝑌))
76opeq2d 4765 . . . 4 (𝑏 = 𝑌 → ⟨(𝐹𝑋), (𝐺𝑏)⟩ = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
8 fvproj.h . . . . 5 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
9 fveq2 6668 . . . . . . 7 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
109opeq1d 4764 . . . . . 6 (𝑥 = 𝑎 → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑦)⟩)
11 fveq2 6668 . . . . . . 7 (𝑦 = 𝑏 → (𝐺𝑦) = (𝐺𝑏))
1211opeq2d 4765 . . . . . 6 (𝑦 = 𝑏 → ⟨(𝐹𝑎), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑏)⟩)
1310, 12cbvmpov 7257 . . . . 5 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = (𝑎𝐴, 𝑏𝐵 ↦ ⟨(𝐹𝑎), (𝐺𝑏)⟩)
148, 13eqtri 2761 . . . 4 𝐻 = (𝑎𝐴, 𝑏𝐵 ↦ ⟨(𝐹𝑎), (𝐺𝑏)⟩)
15 opex 5319 . . . 4 ⟨(𝐹𝑋), (𝐺𝑌)⟩ ∈ V
165, 7, 14, 15ovmpo 7319 . . 3 ((𝑋𝐴𝑌𝐵) → (𝑋𝐻𝑌) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
172, 3, 16syl2anc 587 . 2 (𝜑 → (𝑋𝐻𝑌) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
181, 17eqtr3id 2787 1 (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2113  cop 4519  cfv 6333  (class class class)co 7164  cmpo 7166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169
This theorem is referenced by:  fimaproj  7848  ex-fpar  28391  qtophaus  31350
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