Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5071 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
2 | cofunexg 6002 | . . . 4 ⊢ ((Fun ◡𝐵 ∧ ◡𝐴 ∈ V) → (◡𝐵 ∘ ◡𝐴) ∈ V) | |
3 | 1, 2 | sylan2 284 | . . 3 ⊢ ((Fun ◡𝐵 ∧ 𝐴 ∈ 𝑉) → (◡𝐵 ∘ ◡𝐴) ∈ V) |
4 | cnvco 4719 | . . . . 5 ⊢ ◡(◡𝐵 ∘ ◡𝐴) = (◡◡𝐴 ∘ ◡◡𝐵) | |
5 | cocnvcnv2 5045 | . . . . 5 ⊢ (◡◡𝐴 ∘ ◡◡𝐵) = (◡◡𝐴 ∘ 𝐵) | |
6 | cocnvcnv1 5044 | . . . . 5 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) | |
7 | 4, 5, 6 | 3eqtrri 2163 | . . . 4 ⊢ (𝐴 ∘ 𝐵) = ◡(◡𝐵 ∘ ◡𝐴) |
8 | cnvexg 5071 | . . . 4 ⊢ ((◡𝐵 ∘ ◡𝐴) ∈ V → ◡(◡𝐵 ∘ ◡𝐴) ∈ V) | |
9 | 7, 8 | eqeltrid 2224 | . . 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 1480 Vcvv 2681 ◡ccnv 4533 ∘ ccom 4538 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: (None) |
Copyright terms: Public domain | W3C validator |