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Mirrors > Home > MPE Home > Th. List > mpofun | Structured version Visualization version GIF version |
Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.) (Proof shortened by SN, 23-Jul-2024.) |
Ref | Expression |
---|---|
mpofun.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
mpofun | ⊢ Fun 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeq 3702 | . . . 4 ⊢ ∃*𝑧 𝑧 = 𝐶 | |
2 | 1 | moani 2547 | . . 3 ⊢ ∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) |
3 | 2 | funoprab 7526 | . 2 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
4 | mpofun.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
5 | df-mpo 7410 | . . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
6 | 4, 5 | eqtri 2760 | . . 3 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
7 | 6 | funeqi 6566 | . 2 ⊢ (Fun 𝐹 ↔ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}) |
8 | 3, 7 | mpbir 230 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 Fun wfun 6534 {coprab 7406 ∈ cmpo 7407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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 df-oprab 7409 df-mpo 7410 |
This theorem is referenced by: ofexg 7671 mpoexxg 8058 mpoexw 8061 mpocurryd 8250 imasvscafn 17479 coapm 18017 oppglsm 19504 gsum2d2lem 19835 evlslem2 21633 xkococnlem 23154 ucnima 23777 ucnprima 23778 fmucnd 23788 scutf 27302 smatrcl 32764 smatlem 32765 txomap 32802 tpr2rico 32880 elunirnmbfm 33238 relowlpssretop 36233 aovmpt4g 45895 mpoexxg2 46966 |
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