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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnopabco | Structured version Visualization version GIF version |
Description: Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
fnopabco.1 | ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶) |
fnopabco.2 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
fnopabco.3 | ⊢ 𝐺 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))} |
Ref | Expression |
---|---|
fnopabco | ⊢ (𝐻 Fn 𝐶 → 𝐺 = (𝐻 ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnopabco.3 | . . 3 ⊢ 𝐺 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))} | |
2 | df-mpt 5158 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))} | |
3 | 1, 2 | eqtr4i 2769 | . 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)) |
4 | fnopabco.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶) | |
5 | 4 | adantl 482 | . . 3 ⊢ ((𝐻 Fn 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
6 | fnopabco.2 | . . . . 5 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
7 | df-mpt 5158 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
8 | 6, 7 | eqtr4i 2769 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
9 | 8 | a1i 11 | . . 3 ⊢ (𝐻 Fn 𝐶 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
10 | dffn5 6828 | . . . 4 ⊢ (𝐻 Fn 𝐶 ↔ 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦))) | |
11 | 10 | biimpi 215 | . . 3 ⊢ (𝐻 Fn 𝐶 → 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦))) |
12 | fveq2 6774 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐻‘𝑦) = (𝐻‘𝐵)) | |
13 | 5, 9, 11, 12 | fmptco 7001 | . 2 ⊢ (𝐻 Fn 𝐶 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵))) |
14 | 3, 13 | eqtr4id 2797 | 1 ⊢ (𝐻 Fn 𝐶 → 𝐺 = (𝐻 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {copab 5136 ↦ cmpt 5157 ∘ ccom 5593 Fn wfn 6428 ‘cfv 6433 |
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-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 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-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 |
This theorem is referenced by: opropabco 35882 |
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