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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 6560 | . . . . 5 ⊢ (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧) | |
2 | funmo 6560 | . . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦) | |
3 | 2 | alrimiv 1931 | . . . . 5 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦) |
4 | moexexvw 2625 | . . . . 5 ⊢ ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) | |
5 | 1, 3, 4 | syl2anr 598 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) |
6 | 5 | alrimiv 1931 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) |
7 | funopab 6580 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) | |
8 | 6, 7 | sylibr 233 | . 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}) |
9 | df-co 5684 | . . 3 ⊢ (𝐹 ∘ 𝐺) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} | |
10 | 9 | funeqi 6566 | . 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}) |
11 | 8, 10 | sylibr 233 | 1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 ∃wex 1782 ∃*wmo 2533 class class class wbr 5147 {copab 5209 ∘ ccom 5679 Fun wfun 6534 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-fun 6542 |
This theorem is referenced by: funresfunco 6586 fncofn 6663 fncoOLD 6665 fco3OLD 6748 f1cof1 6795 f1coOLD 6797 curry1 8085 curry2 8088 tposfun 8222 fsuppco 9393 fsuppco2 9394 fsuppcor 9395 fin23lem30 10333 smobeth 10577 hashkf 14288 precsexlem10 27642 precsexlem11 27643 xppreima 31849 smatrcl 32714 comptiunov2i 42390 hoicvr 45199 |
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