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Mirrors > Home > MPE Home > Th. List > funco | Structured version Visualization version GIF version |
Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
funco | ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmo 6364 | . . . . 5 ⊢ (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧) | |
2 | funmo 6364 | . . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦) | |
3 | 2 | alrimiv 1919 | . . . . 5 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦) |
4 | moexexvw 2706 | . . . . 5 ⊢ ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) | |
5 | 1, 3, 4 | syl2anr 596 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) |
6 | 5 | alrimiv 1919 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) |
7 | funopab 6383 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) | |
8 | 6, 7 | sylibr 235 | . 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}) |
9 | df-co 5557 | . . 3 ⊢ (𝐹 ∘ 𝐺) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} | |
10 | 9 | funeqi 6369 | . 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}) |
11 | 8, 10 | sylibr 235 | 1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 ∃wex 1771 ∃*wmo 2613 class class class wbr 5057 {copab 5119 ∘ ccom 5552 Fun wfun 6342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-fun 6350 |
This theorem is referenced by: funresfunco 6389 fnco 6458 f1co 6578 curry1 7788 curry2 7791 tposfun 7897 fsuppco 8853 fsuppco2 8854 fsuppcor 8855 fin23lem30 9752 smobeth 9996 hashkf 13680 xppreima 30322 smatrcl 30960 comptiunov2i 39929 fco3 41367 hoicvr 42707 |
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