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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 5820 | . . . . 5 ⊢ (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧) | |
2 | funmo 5820 | . . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦) | |
3 | 2 | alrimiv 1842 | . . . . 5 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦) |
4 | moexexv 2530 | . . . . 5 ⊢ ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) | |
5 | 1, 3, 4 | syl2anr 494 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) |
6 | 5 | alrimiv 1842 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) |
7 | funopab 5837 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)) | |
8 | 6, 7 | sylibr 223 | . 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}) |
9 | df-co 5047 | . . 3 ⊢ (𝐹 ∘ 𝐺) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} | |
10 | 9 | funeqi 5824 | . 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ Fun {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}) |
11 | 8, 10 | sylibr 223 | 1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 ∃*wmo 2459 class class class wbr 4583 {copab 4642 ∘ ccom 5042 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-fun 5806 |
This theorem is referenced by: fnco 5913 f1co 6023 curry1 7156 curry2 7159 tposfun 7255 fsuppco 8190 fsuppco2 8191 fsuppcor 8192 fin23lem30 9047 smobeth 9287 hashkf 12981 xppreima 28829 smatrcl 29190 comptiunov2i 37017 fco3 38416 hoicvr 39438 funresfunco 39854 |
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