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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 4032 | . . . 4 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V) | |
2 | df-rel 5543 | . . . 4 ⊢ (Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V)) | |
3 | 1, 2 | mpbir 234 | . . 3 ⊢ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) |
4 | df-image 33852 | . . . 4 ⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) | |
5 | 4 | releqi 5634 | . . 3 ⊢ (Rel Image𝐴 ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))) |
6 | 3, 5 | mpbir 234 | . 2 ⊢ Rel Image𝐴 |
7 | vex 3402 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | vex 3402 | . . . . . 6 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brimage 33914 | . . . . 5 ⊢ (𝑥Image𝐴𝑦 ↔ 𝑦 = (𝐴 “ 𝑥)) |
10 | vex 3402 | . . . . . 6 ⊢ 𝑧 ∈ V | |
11 | 7, 10 | brimage 33914 | . . . . 5 ⊢ (𝑥Image𝐴𝑧 ↔ 𝑧 = (𝐴 “ 𝑥)) |
12 | eqtr3 2758 | . . . . 5 ⊢ ((𝑦 = (𝐴 “ 𝑥) ∧ 𝑧 = (𝐴 “ 𝑥)) → 𝑦 = 𝑧) | |
13 | 9, 11, 12 | syl2anb 601 | . . . 4 ⊢ ((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧) |
14 | 13 | gen2 1804 | . . 3 ⊢ ∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧) |
15 | 14 | ax-gen 1803 | . 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧) |
16 | dffun2 6368 | . 2 ⊢ (Fun Image𝐴 ↔ (Rel Image𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧))) | |
17 | 6, 15, 16 | mpbir2an 711 | 1 ⊢ Fun Image𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 = wceq 1543 Vcvv 3398 ∖ cdif 3850 ⊆ wss 3853 △ csymdif 4142 class class class wbr 5039 E cep 5444 × cxp 5534 ◡ccnv 5535 ran crn 5537 “ cima 5539 ∘ ccom 5540 Rel wrel 5541 Fun wfun 6352 ⊗ ctxp 33818 Imagecimage 33828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-symdif 4143 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-eprel 5445 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-1st 7739 df-2nd 7740 df-txp 33842 df-image 33852 |
This theorem is referenced by: fnimage 33917 imageval 33918 imagesset 33941 |
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