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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 5121 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))} | |
3 | 1, 2 | eqtr4i 2765 | . 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)) |
4 | fnopabco.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶) | |
5 | 4 | adantl 485 | . . 3 ⊢ ((𝐻 Fn 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
6 | fnopabco.2 | . . . . 5 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
7 | df-mpt 5121 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
8 | 6, 7 | eqtr4i 2765 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
9 | 8 | a1i 11 | . . 3 ⊢ (𝐻 Fn 𝐶 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
10 | dffn5 6740 | . . . 4 ⊢ (𝐻 Fn 𝐶 ↔ 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦))) | |
11 | 10 | biimpi 219 | . . 3 ⊢ (𝐻 Fn 𝐶 → 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦))) |
12 | fveq2 6686 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐻‘𝑦) = (𝐻‘𝐵)) | |
13 | 5, 9, 11, 12 | fmptco 6913 | . 2 ⊢ (𝐻 Fn 𝐶 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵))) |
14 | 3, 13 | eqtr4id 2793 | 1 ⊢ (𝐻 Fn 𝐶 → 𝐺 = (𝐻 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {copab 5102 ↦ cmpt 5120 ∘ ccom 5539 Fn wfn 6344 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 |
This theorem is referenced by: opropabco 35537 |
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