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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 6143 | . . . . 5 ⊢ (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧) | |
2 | funmo 6143 | . . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦) | |
3 | 2 | alrimiv 2026 | . . . . 5 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦) |
4 | moexexv 2722 | . . . . 5 ⊢ ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) | |
5 | 1, 3, 4 | syl2anr 590 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) |
6 | 5 | alrimiv 2026 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) |
7 | funopab 6162 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) | |
8 | 6, 7 | sylibr 226 | . 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}) |
9 | df-co 5355 | . . 3 ⊢ (𝐹 ∘ 𝐺) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} | |
10 | 9 | funeqi 6148 | . 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}) |
11 | 8, 10 | sylibr 226 | 1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∀wal 1654 ∃wex 1878 ∃*wmo 2603 class class class wbr 4875 {copab 4937 ∘ ccom 5350 Fun wfun 6121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-fun 6129 |
This theorem is referenced by: funresfunco 6168 fnco 6236 f1co 6352 curry1 7538 curry2 7541 tposfun 7638 fsuppco 8582 fsuppco2 8583 fsuppcor 8584 fin23lem30 9486 smobeth 9730 hashkf 13419 xppreima 29994 smatrcl 30403 comptiunov2i 38834 fco3 40221 hoicvr 41550 |
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