Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > funimage | Structured version Visualization version GIF version |
Description: Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
funimage | ⊢ Fun Image𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4105 | . . . 4 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V) | |
2 | df-rel 5555 | . . . 4 ⊢ (Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V)) | |
3 | 1, 2 | mpbir 232 | . . 3 ⊢ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) |
4 | df-image 33222 | . . . 4 ⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) | |
5 | 4 | releqi 5645 | . . 3 ⊢ (Rel Image𝐴 ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))) |
6 | 3, 5 | mpbir 232 | . 2 ⊢ Rel Image𝐴 |
7 | vex 3495 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | vex 3495 | . . . . . 6 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brimage 33284 | . . . . 5 ⊢ (𝑥Image𝐴𝑦 ↔ 𝑦 = (𝐴 “ 𝑥)) |
10 | vex 3495 | . . . . . 6 ⊢ 𝑧 ∈ V | |
11 | 7, 10 | brimage 33284 | . . . . 5 ⊢ (𝑥Image𝐴𝑧 ↔ 𝑧 = (𝐴 “ 𝑥)) |
12 | eqtr3 2840 | . . . . 5 ⊢ ((𝑦 = (𝐴 “ 𝑥) ∧ 𝑧 = (𝐴 “ 𝑥)) → 𝑦 = 𝑧) | |
13 | 9, 11, 12 | syl2anb 597 | . . . 4 ⊢ ((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧) |
14 | 13 | gen2 1788 | . . 3 ⊢ ∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧) |
15 | 14 | ax-gen 1787 | . 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧) |
16 | dffun2 6358 | . 2 ⊢ (Fun Image𝐴 ↔ (Rel Image𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧))) | |
17 | 6, 15, 16 | mpbir2an 707 | 1 ⊢ Fun Image𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 = wceq 1528 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 △ csymdif 4215 class class class wbr 5057 E cep 5457 × cxp 5546 ◡ccnv 5547 ran crn 5549 “ cima 5551 ∘ ccom 5552 Rel wrel 5553 Fun wfun 6342 ⊗ ctxp 33188 Imagecimage 33198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-symdif 4216 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-eprel 5458 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7678 df-2nd 7679 df-txp 33212 df-image 33222 |
This theorem is referenced by: fnimage 33287 imageval 33288 imagesset 33311 |
Copyright terms: Public domain | W3C validator |