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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 4833 | . . . . . 6 ⊢ (𝑋 ∈ ran 𝐹 → ({𝑋} = {𝑌} → 𝑋 = 𝑌)) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌)) |
5 | 4 | necon3d 2953 | . . . 4 ⊢ (𝜑 → (𝑋 ≠ 𝑌 → {𝑋} ≠ {𝑌})) |
6 | 1, 5 | mpd 15 | . . 3 ⊢ (𝜑 → {𝑋} ≠ {𝑌}) |
7 | preimane.f | . . . . 5 ⊢ (𝜑 → Fun 𝐹) | |
8 | funimacnv 6620 | . . . . 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 4805 | . . . . 5 ⊢ (𝜑 → {𝑋} ⊆ ran 𝐹) |
11 | df-ss 3958 | . . . . 5 ⊢ ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋}) | |
12 | 10, 11 | sylib 217 | . . . 4 ⊢ (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋}) |
13 | 9, 12 | eqtrd 2764 | . . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) = {𝑋}) |
14 | funimacnv 6620 | . . . . 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 4805 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ ran 𝐹) |
18 | df-ss 3958 | . . . . 5 ⊢ ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌}) | |
19 | 17, 18 | sylib 217 | . . . 4 ⊢ (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌}) |
20 | 15, 19 | eqtrd 2764 | . . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑌})) = {𝑌}) |
21 | 6, 13, 20 | 3netr4d 3010 | . 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑋})) ≠ (𝐹 “ (◡𝐹 “ {𝑌}))) |
22 | imaeq2 6046 | . . 3 ⊢ ((◡𝐹 “ {𝑋}) = (◡𝐹 “ {𝑌}) → (𝐹 “ (◡𝐹 “ {𝑋})) = (𝐹 “ (◡𝐹 “ {𝑌}))) | |
23 | 22 | necon3i 2965 | . 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 1533 ∈ wcel 2098 ≠ wne 2932 ∩ cin 3940 ⊆ wss 3941 {csn 4621 ◡ccnv 5666 ran crn 5668 “ cima 5670 Fun wfun 6528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-fun 6536 |
This theorem is referenced by: fnpreimac 32368 |
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