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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > preimane | Structured version Visualization version GIF version |
Description: Different elements have different preimages. (Contributed by Thierry Arnoux, 7-May-2023.) |
Ref | Expression |
---|---|
preimane.f | ⊢ (𝜑 → Fun 𝐹) |
preimane.x | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
preimane.y | ⊢ (𝜑 → 𝑋 ∈ ran 𝐹) |
preimane.1 | ⊢ (𝜑 → 𝑌 ∈ ran 𝐹) |
Ref | Expression |
---|---|
preimane | ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preimane.x | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
2 | preimane.y | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ran 𝐹) | |
3 | sneqrg 4641 | . . . . . 6 ⊢ (𝑋 ∈ ran 𝐹 → ({𝑋} = {𝑌} → 𝑋 = 𝑌)) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌)) |
5 | 4 | necon3d 2983 | . . . 4 ⊢ (𝜑 → (𝑋 ≠ 𝑌 → {𝑋} ≠ {𝑌})) |
6 | 1, 5 | mpd 15 | . . 3 ⊢ (𝜑 → {𝑋} ≠ {𝑌}) |
7 | preimane.f | . . . . 5 ⊢ (𝜑 → Fun 𝐹) | |
8 | funimacnv 6266 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹)) |
10 | 2 | snssd 4613 | . . . . 5 ⊢ (𝜑 → {𝑋} ⊆ ran 𝐹) |
11 | df-ss 3838 | . . . . 5 ⊢ ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋}) | |
12 | 10, 11 | sylib 210 | . . . 4 ⊢ (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋}) |
13 | 9, 12 | eqtrd 2809 | . . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = {𝑋}) |
14 | funimacnv 6266 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑌})) = ({𝑌} ∩ ran 𝐹)) | |
15 | 7, 14 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = ({𝑌} ∩ ran 𝐹)) |
16 | preimane.1 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ran 𝐹) | |
17 | 16 | snssd 4613 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ ran 𝐹) |
18 | df-ss 3838 | . . . . 5 ⊢ ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌}) | |
19 | 17, 18 | sylib 210 | . . . 4 ⊢ (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌}) |
20 | 15, 19 | eqtrd 2809 | . . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = {𝑌}) |
21 | 6, 13, 20 | 3netr4d 3039 | . 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌}))) |
22 | imaeq2 5764 | . . 3 ⊢ ((◡𝐹 “ {𝑋}) = (◡𝐹 “ {𝑌}) → (𝐹 “ (◡𝐹 “ {𝑋})) = (𝐹 “ (◡𝐹 “ {𝑌}))) | |
23 | 22 | necon3i 2994 | . 2 ⊢ ((𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌})) → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌})) |
24 | 21, 23 | syl 17 | 1 ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 ∩ cin 3823 ⊆ wss 3824 {csn 4436 ◡ccnv 5403 ran crn 5405 “ cima 5407 Fun wfun 6180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-br 4927 df-opab 4989 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-fun 6188 |
This theorem is referenced by: fnpreimac 30196 |
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