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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvproj | Structured version Visualization version GIF version |
Description: Value of a function on pairs, given two projections 𝐹 and 𝐺. (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 6552 | . 2 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
2 | fvproj.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
3 | fvproj.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | fveq2 6103 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘𝑎) = (𝐹‘𝑋)) | |
5 | 4 | opeq1d 4346 | . . . 4 ⊢ (𝑎 = 𝑋 → 〈(𝐹‘𝑎), (𝐺‘𝑏)〉 = 〈(𝐹‘𝑋), (𝐺‘𝑏)〉) |
6 | fveq2 6103 | . . . . 5 ⊢ (𝑏 = 𝑌 → (𝐺‘𝑏) = (𝐺‘𝑌)) | |
7 | 6 | opeq2d 4347 | . . . 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 6103 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) | |
10 | 9 | opeq1d 4346 | . . . . . 6 ⊢ (𝑥 = 𝑎 → 〈(𝐹‘𝑥), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑦)〉) |
11 | fveq2 6103 | . . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝐺‘𝑦) = (𝐺‘𝑏)) | |
12 | 11 | opeq2d 4347 | . . . . . 6 ⊢ (𝑦 = 𝑏 → 〈(𝐹‘𝑎), (𝐺‘𝑦)〉 = 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
13 | 10, 12 | cbvmpt2v 6633 | . . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
14 | 8, 13 | eqtri 2632 | . . . 4 ⊢ 𝐻 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑏)〉) |
15 | opex 4859 | . . . 4 ⊢ 〈(𝐹‘𝑋), (𝐺‘𝑌)〉 ∈ V | |
16 | 5, 7, 14, 15 | ovmpt2 6694 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐻𝑌) = 〈(𝐹‘𝑋), (𝐺‘𝑌)〉) |
17 | 2, 3, 16 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = 〈(𝐹‘𝑋), (𝐺‘𝑌)〉) |
18 | 1, 17 | syl5eqr 2658 | 1 ⊢ (𝜑 → (𝐻‘〈𝑋, 𝑌〉) = 〈(𝐹‘𝑋), (𝐺‘𝑌)〉) |
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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