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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 7862 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) | |
2 | cofunexg 7882 | . . . 4 ⊢ ((Fun ◡𝐵 ∧ ◡𝐴 ∈ V) → (◡𝐵 ∘ ◡𝐴) ∈ V) | |
3 | 1, 2 | sylan2 594 | . . 3 ⊢ ((Fun ◡𝐵 ∧ 𝐴 ∈ 𝑉) → (◡𝐵 ∘ ◡𝐴) ∈ V) |
4 | cnvco 5842 | . . . . 5 ⊢ ◡(◡𝐵 ∘ ◡𝐴) = (◡◡𝐴 ∘ ◡◡𝐵) | |
5 | cocnvcnv2 6211 | . . . . 5 ⊢ (◡◡𝐴 ∘ ◡◡𝐵) = (◡◡𝐴 ∘ 𝐵) | |
6 | cocnvcnv1 6210 | . . . . 5 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) | |
7 | 4, 5, 6 | 3eqtrri 2766 | . . . 4 ⊢ (𝐴 ∘ 𝐵) = ◡(◡𝐵 ∘ ◡𝐴) |
8 | cnvexg 7862 | . . . 4 ⊢ ((◡𝐵 ∘ ◡𝐴) ∈ V → ◡(◡𝐵 ∘ ◡𝐴) ∈ V) | |
9 | 7, 8 | eqeltrid 2838 | . . 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 5633 ∘ ccom 5638 Fun wfun 6491 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 |
This theorem is referenced by: fsuppco 9343 |
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