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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 2734 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
4 | 1, 2, 3 | coafval 18117 | . . 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 8089 | . 2 ⊢ dom · ⊆ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) |
6 | iunss 5049 | . . 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 4812 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → {𝑔} ⊆ 𝐴) | |
8 | ssrab2 4089 | . . . 4 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴 | |
9 | xpss12 5703 | . . . 4 ⊢ (({𝑔} ⊆ 𝐴 ∧ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴) → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) | |
10 | 7, 8, 9 | sylancl 586 | . . 3 ⊢ (𝑔 ∈ 𝐴 → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) |
11 | 6, 10 | mprgbir 3065 | . 2 ⊢ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴) |
12 | 5, 11 | sstri 4004 | 1 ⊢ dom · ⊆ (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 {crab 3432 ⊆ wss 3962 {csn 4630 〈cop 4636 〈cotp 4638 ∪ ciun 4995 × cxp 5686 dom cdm 5688 ‘cfv 6562 (class class class)co 7430 2nd c2nd 8011 compcco 17309 domacdoma 18073 codaccoda 18074 Arrowcarw 18075 compaccoa 18107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-ot 4639 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-arw 18080 df-coa 18109 |
This theorem is referenced by: coapm 18124 |
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