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 4770 | . . . . . 6 ⊢ (𝑋 ∈ ran 𝐹 → ({𝑋} = {𝑌} → 𝑋 = 𝑌)) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌)) |
5 | 4 | necon3d 3037 | . . . 4 ⊢ (𝜑 → (𝑋 ≠ 𝑌 → {𝑋} ≠ {𝑌})) |
6 | 1, 5 | mpd 15 | . . 3 ⊢ (𝜑 → {𝑋} ≠ {𝑌}) |
7 | preimane.f | . . . . 5 ⊢ (𝜑 → Fun 𝐹) | |
8 | funimacnv 6435 | . . . . 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 4742 | . . . . 5 ⊢ (𝜑 → {𝑋} ⊆ ran 𝐹) |
11 | df-ss 3952 | . . . . 5 ⊢ ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋}) | |
12 | 10, 11 | sylib 220 | . . . 4 ⊢ (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋}) |
13 | 9, 12 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = {𝑋}) |
14 | funimacnv 6435 | . . . . 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 4742 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ ran 𝐹) |
18 | df-ss 3952 | . . . . 5 ⊢ ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌}) | |
19 | 17, 18 | sylib 220 | . . . 4 ⊢ (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌}) |
20 | 15, 19 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = {𝑌}) |
21 | 6, 13, 20 | 3netr4d 3093 | . 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌}))) |
22 | imaeq2 5925 | . . 3 ⊢ ((◡𝐹 “ {𝑋}) = (◡𝐹 “ {𝑌}) → (𝐹 “ (◡𝐹 “ {𝑋})) = (𝐹 “ (◡𝐹 “ {𝑌}))) | |
23 | 22 | necon3i 3048 | . 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 1537 ∈ wcel 2114 ≠ wne 3016 ∩ cin 3935 ⊆ wss 3936 {csn 4567 ◡ccnv 5554 ran crn 5556 “ cima 5558 Fun wfun 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fun 6357 |
This theorem is referenced by: fnpreimac 30416 |
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