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Mirrors > Home > MPE Home > Th. List > cofunex2g | Structured version Visualization version 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 7852 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
2 | cofunexg 7872 | . . . 4 ⊢ ((Fun ◡𝐵 ∧ ◡𝐴 ∈ V) → (◡𝐵 ∘ ◡𝐴) ∈ V) | |
3 | 1, 2 | sylan2 594 | . . 3 ⊢ ((Fun ◡𝐵 ∧ 𝐴 ∈ 𝑉) → (◡𝐵 ∘ ◡𝐴) ∈ V) |
4 | cnvco 5838 | . . . . 5 ⊢ ◡(◡𝐵 ∘ ◡𝐴) = (◡◡𝐴 ∘ ◡◡𝐵) | |
5 | cocnvcnv2 6207 | . . . . 5 ⊢ (◡◡𝐴 ∘ ◡◡𝐵) = (◡◡𝐴 ∘ 𝐵) | |
6 | cocnvcnv1 6206 | . . . . 5 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) | |
7 | 4, 5, 6 | 3eqtrri 2771 | . . . 4 ⊢ (𝐴 ∘ 𝐵) = ◡(◡𝐵 ∘ ◡𝐴) |
8 | cnvexg 7852 | . . . 4 ⊢ ((◡𝐵 ∘ ◡𝐴) ∈ V → ◡(◡𝐵 ∘ ◡𝐴) ∈ V) | |
9 | 7, 8 | eqeltrid 2843 | . . 3 ⊢ ((◡𝐵 ∘ ◡𝐴) ∈ V → (𝐴 ∘ 𝐵) ∈ V) |
10 | 3, 9 | syl 17 | . 2 ⊢ ((Fun ◡𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∘ 𝐵) ∈ V) |
11 | 10 | ancoms 460 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ Fun ◡𝐵) → (𝐴 ∘ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3444 ◡ccnv 5630 ∘ ccom 5635 Fun wfun 6486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 |
This theorem is referenced by: fsuppco 9272 |
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