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Mirrors > Home > MPE Home > Th. List > dmcoass | Structured version Visualization version GIF version |
Description: The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
coafval.o | ⊢ · = (compa‘𝐶) |
coafval.a | ⊢ 𝐴 = (Arrow‘𝐶) |
Ref | Expression |
---|---|
dmcoass | ⊢ dom · ⊆ (𝐴 × 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coafval.o | . . . 4 ⊢ · = (compa‘𝐶) | |
2 | coafval.a | . . . 4 ⊢ 𝐴 = (Arrow‘𝐶) | |
3 | eqid 2798 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
4 | 1, 2, 3 | coafval 17316 | . . 3 ⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ 〈(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(〈(doma‘𝑓), (doma‘𝑔)〉(comp‘𝐶)(coda‘𝑔))(2nd ‘𝑓))〉) |
5 | 4 | dmmpossx 7746 | . 2 ⊢ dom · ⊆ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) |
6 | iunss 4932 | . . 3 ⊢ (∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴) ↔ ∀𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) | |
7 | snssi 4701 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → {𝑔} ⊆ 𝐴) | |
8 | ssrab2 4007 | . . . 4 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴 | |
9 | xpss12 5534 | . . . 4 ⊢ (({𝑔} ⊆ 𝐴 ∧ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴) → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) | |
10 | 7, 8, 9 | sylancl 589 | . . 3 ⊢ (𝑔 ∈ 𝐴 → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) |
11 | 6, 10 | mprgbir 3121 | . 2 ⊢ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴) |
12 | 5, 11 | sstri 3924 | 1 ⊢ dom · ⊆ (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 {csn 4525 〈cop 4531 〈cotp 4533 ∪ ciun 4881 × cxp 5517 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 2nd c2nd 7670 compcco 16569 domacdoma 17272 codaccoda 17273 Arrowcarw 17274 compaccoa 17306 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-arw 17279 df-coa 17308 |
This theorem is referenced by: coapm 17323 |
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