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Mirrors > Home > MPE Home > Th. List > fvproj | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fvproj.h | ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) |
fvproj.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
fvproj.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
fvproj | ⊢ (𝜑 → (𝐻‘〈𝑋, 𝑌〉) = 〈(𝐹‘𝑋), (𝐺‘𝑌)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7434 | . 2 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
2 | fvproj.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
3 | fvproj.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | fveq2 6907 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘𝑎) = (𝐹‘𝑋)) | |
5 | 4 | opeq1d 4884 | . . . 4 ⊢ (𝑎 = 𝑋 → 〈(𝐹‘𝑎), (𝐺‘𝑏)〉 = 〈(𝐹‘𝑋), (𝐺‘𝑏)〉) |
6 | fveq2 6907 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝐺‘𝑏) = (𝐺‘𝑌)) | |
7 | 6 | opeq2d 4885 | . . . 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 6907 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) | |
10 | 9 | opeq1d 4884 | . . . . . 6 ⊢ (𝑥 = 𝑎 → 〈(𝐹‘𝑥), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑦)〉) |
11 | fveq2 6907 | . . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝐺‘𝑦) = (𝐺‘𝑏)) | |
12 | 11 | opeq2d 4885 | . . . . . 6 ⊢ (𝑦 = 𝑏 → 〈(𝐹‘𝑎), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
13 | 10, 12 | cbvmpov 7528 | . . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
14 | 8, 13 | eqtri 2763 | . . . 4 ⊢ 𝐻 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
15 | opex 5475 | . . . 4 ⊢ 〈(𝐹‘𝑋), (𝐺‘𝑌)〉 ∈ V | |
16 | 5, 7, 14, 15 | ovmpo 7593 | . . 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 2789 | 1 ⊢ (𝜑 → (𝐻‘〈𝑋, 𝑌〉) = 〈(𝐹‘𝑋), (𝐺‘𝑌)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 〈cop 4637 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: fimaproj 8159 ex-fpar 30491 qtophaus 33797 |
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