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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 2731 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
4 | 1, 2, 3 | coafval 18024 | . . 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 8056 | . 2 ⊢ dom · ⊆ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) |
6 | iunss 5048 | . . 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 4811 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → {𝑔} ⊆ 𝐴) | |
8 | ssrab2 4077 | . . . 4 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴 | |
9 | xpss12 5691 | . . . 4 ⊢ (({𝑔} ⊆ 𝐴 ∧ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴) → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) | |
10 | 7, 8, 9 | sylancl 585 | . . 3 ⊢ (𝑔 ∈ 𝐴 → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) |
11 | 6, 10 | mprgbir 3067 | . 2 ⊢ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴) |
12 | 5, 11 | sstri 3991 | 1 ⊢ dom · ⊆ (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 {crab 3431 ⊆ wss 3948 {csn 4628 〈cop 4634 〈cotp 4636 ∪ ciun 4997 × cxp 5674 dom cdm 5676 ‘cfv 6543 (class class class)co 7412 2nd c2nd 7978 compcco 17216 domacdoma 17980 codaccoda 17981 Arrowcarw 17982 compaccoa 18014 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-ot 4637 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-arw 17987 df-coa 18016 |
This theorem is referenced by: coapm 18031 |
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