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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 7153 | . 2 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
2 | fvproj.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
3 | fvproj.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | fveq2 6664 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘𝑎) = (𝐹‘𝑋)) | |
5 | 4 | opeq1d 4802 | . . . 4 ⊢ (𝑎 = 𝑋 → 〈(𝐹‘𝑎), (𝐺‘𝑏)〉 = 〈(𝐹‘𝑋), (𝐺‘𝑏)〉) |
6 | fveq2 6664 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝐺‘𝑏) = (𝐺‘𝑌)) | |
7 | 6 | opeq2d 4803 | . . . 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 6664 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) | |
10 | 9 | opeq1d 4802 | . . . . . 6 ⊢ (𝑥 = 𝑎 → 〈(𝐹‘𝑥), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑦)〉) |
11 | fveq2 6664 | . . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝐺‘𝑦) = (𝐺‘𝑏)) | |
12 | 11 | opeq2d 4803 | . . . . . 6 ⊢ (𝑦 = 𝑏 → 〈(𝐹‘𝑎), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
13 | 10, 12 | cbvmpov 7243 | . . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
14 | 8, 13 | eqtri 2844 | . . . 4 ⊢ 𝐻 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
15 | opex 5348 | . . . 4 ⊢ 〈(𝐹‘𝑋), (𝐺‘𝑌)〉 ∈ V | |
16 | 5, 7, 14, 15 | ovmpo 7304 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐻𝑌) = 〈(𝐹‘𝑋), (𝐺‘𝑌)〉) |
17 | 2, 3, 16 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = 〈(𝐹‘𝑋), (𝐺‘𝑌)〉) |
18 | 1, 17 | syl5eqr 2870 | 1 ⊢ (𝜑 → (𝐻‘〈𝑋, 𝑌〉) = 〈(𝐹‘𝑋), (𝐺‘𝑌)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cop 4566 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 |
This theorem is referenced by: fimaproj 7823 ex-fpar 28235 qtophaus 31095 |
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