Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2823 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
4 | 1, 2, 3 | coafval 17326 | . . 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 7766 | . 2 ⊢ dom · ⊆ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) |
6 | iunss 4971 | . . 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 4743 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → {𝑔} ⊆ 𝐴) | |
8 | ssrab2 4058 | . . . 4 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴 | |
9 | xpss12 5572 | . . . 4 ⊢ (({𝑔} ⊆ 𝐴 ∧ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴) → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) | |
10 | 7, 8, 9 | sylancl 588 | . . 3 ⊢ (𝑔 ∈ 𝐴 → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) |
11 | 6, 10 | mprgbir 3155 | . 2 ⊢ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴) |
12 | 5, 11 | sstri 3978 | 1 ⊢ dom · ⊆ (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {crab 3144 ⊆ wss 3938 {csn 4569 〈cop 4575 〈cotp 4577 ∪ ciun 4921 × cxp 5555 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 2nd c2nd 7690 compcco 16579 domacdoma 17282 codaccoda 17283 Arrowcarw 17284 compaccoa 17316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-ot 4578 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-arw 17289 df-coa 17318 |
This theorem is referenced by: coapm 17333 |
Copyright terms: Public domain | W3C validator |