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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 5032 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relssdmrn 5054 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) |
4 | dmcoss 4803 | . . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | |
5 | dmexg 4798 | . . . . 5 ⊢ (𝐵 ∈ 𝐶 → dom 𝐵 ∈ V) | |
6 | ssexg 4062 | . . . . 5 ⊢ ((dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴 ∘ 𝐵) ∈ V) | |
7 | 4, 5, 6 | sylancr 410 | . . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ∘ 𝐵) ∈ V) |
8 | 7 | adantl 275 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ∘ 𝐵) ∈ V) |
9 | rnco 5040 | . . . 4 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
10 | rnexg 4799 | . . . . . 6 ⊢ (𝐵 ∈ 𝐶 → ran 𝐵 ∈ V) | |
11 | resfunexg 5634 | . . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V) | |
12 | 10, 11 | sylan2 284 | . . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ ran 𝐵) ∈ V) |
13 | rnexg 4799 | . . . . 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 | eqeltrid 2224 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ∘ 𝐵) ∈ V) |
16 | xpexg 4648 | . . 3 ⊢ ((dom (𝐴 ∘ 𝐵) ∈ V ∧ ran (𝐴 ∘ 𝐵) ∈ V) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V) | |
17 | 8, 15, 16 | syl2anc 408 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V) |
18 | ssexg 4062 | . 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∧ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V) → (𝐴 ∘ 𝐵) ∈ V) | |
19 | 3, 17, 18 | sylancr 410 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 × cxp 4532 dom cdm 4534 ran crn 4535 ↾ cres 4536 ∘ ccom 4538 Rel wrel 4539 Fun wfun 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: cofunex2g 6003 ctm 6987 ctssdclemr 6990 |
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