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Theorem fvproj 29884
Description: Value of a function on pairs, given two projections 𝐹 and 𝐺. (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 6650 . 2 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2 fvproj.x . . 3 (𝜑𝑋𝐴)
3 fvproj.y . . 3 (𝜑𝑌𝐵)
4 fveq2 6189 . . . . 5 (𝑎 = 𝑋 → (𝐹𝑎) = (𝐹𝑋))
54opeq1d 4406 . . . 4 (𝑎 = 𝑋 → ⟨(𝐹𝑎), (𝐺𝑏)⟩ = ⟨(𝐹𝑋), (𝐺𝑏)⟩)
6 fveq2 6189 . . . . 5 (𝑏 = 𝑌 → (𝐺𝑏) = (𝐺𝑌))
76opeq2d 4407 . . . 4 (𝑏 = 𝑌 → ⟨(𝐹𝑋), (𝐺𝑏)⟩ = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
8 fvproj.h . . . . 5 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
9 fveq2 6189 . . . . . . 7 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
109opeq1d 4406 . . . . . 6 (𝑥 = 𝑎 → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑦)⟩)
11 fveq2 6189 . . . . . . 7 (𝑦 = 𝑏 → (𝐺𝑦) = (𝐺𝑏))
1211opeq2d 4407 . . . . . 6 (𝑦 = 𝑏 → ⟨(𝐹𝑎), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑏)⟩)
1310, 12cbvmpt2v 6732 . . . . 5 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = (𝑎𝐴, 𝑏𝐵 ↦ ⟨(𝐹𝑎), (𝐺𝑏)⟩)
148, 13eqtri 2643 . . . 4 𝐻 = (𝑎𝐴, 𝑏𝐵 ↦ ⟨(𝐹𝑎), (𝐺𝑏)⟩)
15 opex 4930 . . . 4 ⟨(𝐹𝑋), (𝐺𝑌)⟩ ∈ V
165, 7, 14, 15ovmpt2 6793 . . 3 ((𝑋𝐴𝑌𝐵) → (𝑋𝐻𝑌) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
172, 3, 16syl2anc 693 . 2 (𝜑 → (𝑋𝐻𝑌) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
181, 17syl5eqr 2669 1 (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  cop 4181  cfv 5886  (class class class)co 6647  cmpt2 6649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652
This theorem is referenced by:  fimaproj  29885  qtophaus  29888
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