Theorem fvproj 8115

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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
fvproj.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
fvproj.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
fvproj	⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)

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.

Step	Hyp	Ref	Expression
1		df-ov 7392	. 2 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2		fvproj.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
3		fvproj.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		fveq2 6860	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘𝑎) = (𝐹‘𝑋))
5	4	opeq1d 4845	. . . 4 ⊢ (𝑎 = 𝑋 → ⟨(𝐹‘𝑎), (𝐺‘𝑏)⟩ = ⟨(𝐹‘𝑋), (𝐺‘𝑏)⟩)
6		fveq2 6860	. . . . 5 ⊢ (𝑏 = 𝑌 → (𝐺‘𝑏) = (𝐺‘𝑌))
7	6	opeq2d 4846	. . . 4 ⊢ (𝑏 = 𝑌 → ⟨(𝐹‘𝑋), (𝐺‘𝑏)⟩ = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
8		fvproj.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
9		fveq2 6860	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
10	9	opeq1d 4845	. . . . . 6 ⊢ (𝑥 = 𝑎 → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑦)⟩)
11		fveq2 6860	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝐺‘𝑦) = (𝐺‘𝑏))
12	11	opeq2d 4846	. . . . . 6 ⊢ (𝑦 = 𝑏 → ⟨(𝐹‘𝑎), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑏)⟩)
13	10, 12	cbvmpov 7486	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑏)⟩)
14	8, 13	eqtri 2753	. . . 4 ⊢ 𝐻 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑏)⟩)
15		opex 5426	. . . 4 ⊢ ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩ ∈ V
16	5, 7, 14, 15	ovmpo 7551	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐻𝑌) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
17	2, 3, 16	syl2anc 584	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
18	1, 17	eqtr3id 2779	1 ⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4597  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6515  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394

This theorem is referenced by:  fimaproj  8116  ex-fpar  30397  qtophaus  33832