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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 7278 | . 2 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
2 | fvproj.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
3 | fvproj.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | fveq2 6774 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘𝑎) = (𝐹‘𝑋)) | |
5 | 4 | opeq1d 4810 | . . . 4 ⊢ (𝑎 = 𝑋 → 〈(𝐹‘𝑎), (𝐺‘𝑏)〉 = 〈(𝐹‘𝑋), (𝐺‘𝑏)〉) |
6 | fveq2 6774 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝐺‘𝑏) = (𝐺‘𝑌)) | |
7 | 6 | opeq2d 4811 | . . . 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 6774 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) | |
10 | 9 | opeq1d 4810 | . . . . . 6 ⊢ (𝑥 = 𝑎 → 〈(𝐹‘𝑥), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑦)〉) |
11 | fveq2 6774 | . . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝐺‘𝑦) = (𝐺‘𝑏)) | |
12 | 11 | opeq2d 4811 | . . . . . 6 ⊢ (𝑦 = 𝑏 → 〈(𝐹‘𝑎), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
13 | 10, 12 | cbvmpov 7370 | . . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
14 | 8, 13 | eqtri 2766 | . . . 4 ⊢ 𝐻 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
15 | opex 5379 | . . . 4 ⊢ 〈(𝐹‘𝑋), (𝐺‘𝑌)〉 ∈ V | |
16 | 5, 7, 14, 15 | ovmpo 7433 | . . 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 2792 | 1 ⊢ (𝜑 → (𝐻‘〈𝑋, 𝑌〉) = 〈(𝐹‘𝑋), (𝐺‘𝑌)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 〈cop 4567 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 |
This theorem is referenced by: fimaproj 7976 ex-fpar 28826 qtophaus 31786 |
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