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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 2826 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
4 | 1, 2, 3 | coafval 17067 | . . 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 | dmmpt2ssx 7499 | . 2 ⊢ dom · ⊆ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) |
6 | iunss 4782 | . . 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 4558 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → {𝑔} ⊆ 𝐴) | |
8 | ssrab2 3913 | . . . 4 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴 | |
9 | xpss12 5358 | . . . 4 ⊢ (({𝑔} ⊆ 𝐴 ∧ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴) → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) | |
10 | 7, 8, 9 | sylancl 582 | . . 3 ⊢ (𝑔 ∈ 𝐴 → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)) |
11 | 6, 10 | mprgbir 3137 | . 2 ⊢ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴) |
12 | 5, 11 | sstri 3837 | 1 ⊢ dom · ⊆ (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∈ wcel 2166 {crab 3122 ⊆ wss 3799 {csn 4398 〈cop 4404 〈cotp 4406 ∪ ciun 4741 × cxp 5341 dom cdm 5343 ‘cfv 6124 (class class class)co 6906 2nd c2nd 7428 compcco 16318 domacdoma 17023 codaccoda 17024 Arrowcarw 17025 compaccoa 17057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-ot 4407 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-1st 7429 df-2nd 7430 df-arw 17030 df-coa 17059 |
This theorem is referenced by: coapm 17074 |
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