Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cofunexg | GIF version |
Description: Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.) |
Ref | Expression |
---|---|
cofunexg | ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 4839 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relssdmrn 4861 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵))) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) |
4 | dmcoss 4619 | . . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | |
5 | dmexg 4614 | . . . . 5 ⊢ (𝐵 ∈ 𝐶 → dom 𝐵 ∈ V) | |
6 | ssexg 3917 | . . . . 5 ⊢ ((dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴 ∘ 𝐵) ∈ V) | |
7 | 4, 5, 6 | sylancr 405 | . . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ∘ 𝐵) ∈ V) |
8 | 7 | adantl 271 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ∘ 𝐵) ∈ V) |
9 | rnco 4847 | . . . 4 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
10 | rnexg 4615 | . . . . . 6 ⊢ (𝐵 ∈ 𝐶 → ran 𝐵 ∈ V) | |
11 | resfunexg 5403 | . . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V) | |
12 | 10, 11 | sylan2 280 | . . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ ran 𝐵) ∈ V) |
13 | rnexg 4615 | . . . . 5 ⊢ ((𝐴 ↾ ran 𝐵) ∈ V → ran (𝐴 ↾ ran 𝐵) ∈ V) | |
14 | 12, 13 | syl 14 | . . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ↾ ran 𝐵) ∈ V) |
15 | 9, 14 | syl5eqel 2165 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ∘ 𝐵) ∈ V) |
16 | xpexg 4470 | . . 3 ⊢ ((dom (𝐴 ∘ 𝐵) ∈ V ∧ ran (𝐴 ∘ 𝐵) ∈ V) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V) | |
17 | 8, 15, 16 | syl2anc 403 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V) |
18 | ssexg 3917 | . 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∧ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
19 | 3, 17, 18 | sylancr 405 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 × cxp 4361 dom cdm 4363 ran crn 4364 ↾ cres 4365 ∘ ccom 4367 Rel wrel 4368 Fun wfun 4916 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 |
This theorem is referenced by: cofunex2g 5759 |
Copyright terms: Public domain | W3C validator |