| 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 4839 | . . . . . 6 ⊢ (𝑋 ∈ ran 𝐹 → ({𝑋} = {𝑌} → 𝑋 = 𝑌)) | |
| 4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌)) |
| 5 | 4 | necon3d 2961 | . . . 4 ⊢ (𝜑 → (𝑋 ≠ 𝑌 → {𝑋} ≠ {𝑌})) |
| 6 | 1, 5 | mpd 15 | . . 3 ⊢ (𝜑 → {𝑋} ≠ {𝑌}) |
| 7 | preimane.f | . . . . 5 ⊢ (𝜑 → Fun 𝐹) | |
| 8 | funimacnv 6647 | . . . . 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 4809 | . . . . 5 ⊢ (𝜑 → {𝑋} ⊆ ran 𝐹) |
| 11 | dfss2 3969 | . . . . 5 ⊢ ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋}) | |
| 12 | 10, 11 | sylib 218 | . . . 4 ⊢ (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋}) |
| 13 | 9, 12 | eqtrd 2777 | . . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = {𝑋}) |
| 14 | funimacnv 6647 | . . . . 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 4809 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ ran 𝐹) |
| 18 | dfss2 3969 | . . . . 5 ⊢ ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌}) | |
| 19 | 17, 18 | sylib 218 | . . . 4 ⊢ (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌}) |
| 20 | 15, 19 | eqtrd 2777 | . . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = {𝑌}) |
| 21 | 6, 13, 20 | 3netr4d 3018 | . 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌}))) |
| 22 | imaeq2 6074 | . . 3 ⊢ ((◡𝐹 “ {𝑋}) = (◡𝐹 “ {𝑌}) → (𝐹 “ (◡𝐹 “ {𝑋})) = (𝐹 “ (◡𝐹 “ {𝑌}))) | |
| 23 | 22 | necon3i 2973 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ◡ccnv 5684 ran crn 5686 “ cima 5688 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 |
| This theorem is referenced by: fnpreimac 32681 |
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