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Mirrors > Home > ILE Home > Th. List > cofunex2g | GIF version |
Description: Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.) |
Ref | Expression |
---|---|
cofunex2g | ⊢ ((𝐴 ∈ 𝑉 ∧ Fun ◡𝐵) → (𝐴 ∘ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvexg 5123 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
2 | cofunexg 6059 | . . . 4 ⊢ ((Fun ◡𝐵 ∧ ◡𝐴 ∈ V) → (◡𝐵 ∘ ◡𝐴) ∈ V) | |
3 | 1, 2 | sylan2 284 | . . 3 ⊢ ((Fun ◡𝐵 ∧ 𝐴 ∈ 𝑉) → (◡𝐵 ∘ ◡𝐴) ∈ V) |
4 | cnvco 4771 | . . . . 5 ⊢ ◡(◡𝐵 ∘ ◡𝐴) = (◡◡𝐴 ∘ ◡◡𝐵) | |
5 | cocnvcnv2 5097 | . . . . 5 ⊢ (◡◡𝐴 ∘ ◡◡𝐵) = (◡◡𝐴 ∘ 𝐵) | |
6 | cocnvcnv1 5096 | . . . . 5 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) | |
7 | 4, 5, 6 | 3eqtrri 2183 | . . . 4 ⊢ (𝐴 ∘ 𝐵) = ◡(◡𝐵 ∘ ◡𝐴) |
8 | cnvexg 5123 | . . . 4 ⊢ ((◡𝐵 ∘ ◡𝐴) ∈ V → ◡(◡𝐵 ∘ ◡𝐴) ∈ V) | |
9 | 7, 8 | eqeltrid 2244 | . . 3 ⊢ ((◡𝐵 ∘ ◡𝐴) ∈ V → (𝐴 ∘ 𝐵) ∈ V) |
10 | 3, 9 | syl 14 | . 2 ⊢ ((Fun ◡𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∘ 𝐵) ∈ V) |
11 | 10 | ancoms 266 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ Fun ◡𝐵) → (𝐴 ∘ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2128 Vcvv 2712 ◡ccnv 4585 ∘ ccom 4590 Fun wfun 5164 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 |
This theorem is referenced by: (None) |
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