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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 3699 | . . . 4 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V) | |
2 | df-rel 5045 | . . . 4 ⊢ (Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V)) | |
3 | 1, 2 | mpbir 220 | . . 3 ⊢ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) |
4 | df-image 31140 | . . . 4 ⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) | |
5 | 4 | releqi 5125 | . . 3 ⊢ (Rel Image𝐴 ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))) |
6 | 3, 5 | mpbir 220 | . 2 ⊢ Rel Image𝐴 |
7 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brimage 31203 | . . . . 5 ⊢ (𝑥Image𝐴𝑦 ↔ 𝑦 = (𝐴 “ 𝑥)) |
10 | vex 3176 | . . . . . 6 ⊢ 𝑧 ∈ V | |
11 | 7, 10 | brimage 31203 | . . . . 5 ⊢ (𝑥Image𝐴𝑧 ↔ 𝑧 = (𝐴 “ 𝑥)) |
12 | eqtr3 2631 | . . . . 5 ⊢ ((𝑦 = (𝐴 “ 𝑥) ∧ 𝑧 = (𝐴 “ 𝑥)) → 𝑦 = 𝑧) | |
13 | 9, 11, 12 | syl2anb 495 | . . . 4 ⊢ ((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧) |
14 | 13 | gen2 1714 | . . 3 ⊢ ∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧) |
15 | 14 | ax-gen 1713 | . 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧) |
16 | dffun2 5814 | . 2 ⊢ (Fun Image𝐴 ↔ (Rel Image𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧))) | |
17 | 6, 15, 16 | mpbir2an 957 | 1 ⊢ Fun Image𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 = wceq 1475 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 △ csymdif 3805 class class class wbr 4583 E cep 4947 × cxp 5036 ◡ccnv 5037 ran crn 5039 “ cima 5041 ∘ ccom 5042 Rel wrel 5043 Fun wfun 5798 ⊗ ctxp 31106 Imagecimage 31116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-eprel 4949 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-image 31140 |
This theorem is referenced by: fnimage 31206 imageval 31207 imagesset 31230 |
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