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Theorem fvproj 29227
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 6552 . 2 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2 fvproj.x . . 3 (𝜑𝑋𝐴)
3 fvproj.y . . 3 (𝜑𝑌𝐵)
4 fveq2 6103 . . . . 5 (𝑎 = 𝑋 → (𝐹𝑎) = (𝐹𝑋))
54opeq1d 4346 . . . 4 (𝑎 = 𝑋 → ⟨(𝐹𝑎), (𝐺𝑏)⟩ = ⟨(𝐹𝑋), (𝐺𝑏)⟩)
6 fveq2 6103 . . . . 5 (𝑏 = 𝑌 → (𝐺𝑏) = (𝐺𝑌))
76opeq2d 4347 . . . 4 (𝑏 = 𝑌 → ⟨(𝐹𝑋), (𝐺𝑏)⟩ = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
8 fvproj.h . . . . 5 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
9 fveq2 6103 . . . . . . 7 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
109opeq1d 4346 . . . . . 6 (𝑥 = 𝑎 → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑦)⟩)
11 fveq2 6103 . . . . . . 7 (𝑦 = 𝑏 → (𝐺𝑦) = (𝐺𝑏))
1211opeq2d 4347 . . . . . 6 (𝑦 = 𝑏 → ⟨(𝐹𝑎), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑏)⟩)
1310, 12cbvmpt2v 6633 . . . . 5 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = (𝑎𝐴, 𝑏𝐵 ↦ ⟨(𝐹𝑎), (𝐺𝑏)⟩)
148, 13eqtri 2632 . . . 4 𝐻 = (𝑎𝐴, 𝑏𝐵 ↦ ⟨(𝐹𝑎), (𝐺𝑏)⟩)
15 opex 4859 . . . 4 ⟨(𝐹𝑋), (𝐺𝑌)⟩ ∈ V
165, 7, 14, 15ovmpt2 6694 . . 3 ((𝑋𝐴𝑌𝐵) → (𝑋𝐻𝑌) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
172, 3, 16syl2anc 691 . 2 (𝜑 → (𝑋𝐻𝑌) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
181, 17syl5eqr 2658 1 (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  cop 4131  cfv 5804  (class class class)co 6549  cmpt2 6551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554
This theorem is referenced by:  fimaproj  29228  qtophaus  29231
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