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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.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶) | |
2 | 1 | adantl 484 | . . 3 ⊢ ((𝐻 Fn 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
3 | fnopabco.2 | . . . . 5 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
4 | df-mpt 5149 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
5 | 3, 4 | eqtr4i 2849 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝐻 Fn 𝐶 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
7 | dffn5 6726 | . . . 4 ⊢ (𝐻 Fn 𝐶 ↔ 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦))) | |
8 | 7 | biimpi 218 | . . 3 ⊢ (𝐻 Fn 𝐶 → 𝐻 = (𝑦 ∈ 𝐶 ↦ (𝐻‘𝑦))) |
9 | fveq2 6672 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐻‘𝑦) = (𝐻‘𝐵)) | |
10 | 2, 6, 8, 9 | fmptco 6893 | . 2 ⊢ (𝐻 Fn 𝐶 → (𝐻 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵))) |
11 | fnopabco.3 | . . 3 ⊢ 𝐺 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))} | |
12 | df-mpt 5149 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))} | |
13 | 11, 12 | eqtr4i 2849 | . 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝐵)) |
14 | 10, 13 | syl6reqr 2877 | 1 ⊢ (𝐻 Fn 𝐶 → 𝐺 = (𝐻 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {copab 5130 ↦ cmpt 5148 ∘ ccom 5561 Fn wfn 6352 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 |
This theorem is referenced by: opropabco 35001 |
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