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Mirrors > Home > MPE Home > Th. List > Mathboxes > brfullfun | Structured version Visualization version GIF version |
Description: A binary relation form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brfullfun.1 | ⊢ 𝐴 ∈ V |
brfullfun.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brfullfun | ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2738 | . 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐵 = (FullFun𝐹‘𝐴)) | |
2 | fullfunfnv 35390 | . . 3 ⊢ FullFun𝐹 Fn V | |
3 | brfullfun.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | fnbrfvb 6944 | . . 3 ⊢ ((FullFun𝐹 Fn V ∧ 𝐴 ∈ V) → ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵)) | |
5 | 2, 3, 4 | mp2an 689 | . 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵) |
6 | fullfunfv 35391 | . . 3 ⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴) | |
7 | 6 | eqeq2i 2744 | . 2 ⊢ (𝐵 = (FullFun𝐹‘𝐴) ↔ 𝐵 = (𝐹‘𝐴)) |
8 | 1, 5, 7 | 3bitr3i 301 | 1 ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 ∈ wcel 2105 Vcvv 3473 class class class wbr 5148 Fn wfn 6538 ‘cfv 6543 FullFuncfullfn 35294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-symdif 4242 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7979 df-2nd 7980 df-txp 35298 df-singleton 35306 df-singles 35307 df-image 35308 df-funpart 35318 df-fullfun 35319 |
This theorem is referenced by: dfrecs2 35394 dfrdg4 35395 |
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