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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1resrcmplf1dlem | Structured version Visualization version GIF version |
Description: Lemma for f1resrcmplf1d 33682. (Contributed by BTernaryTau, 27-Sep-2023.) |
Ref | Expression |
---|---|
f1resrcmplf1dlem.1 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
f1resrcmplf1dlem.2 | ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
f1resrcmplf1dlem.3 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
f1resrcmplf1dlem.4 | ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅) |
Ref | Expression |
---|---|
f1resrcmplf1dlem | ⊢ (𝜑 → ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1resrcmplf1dlem.1 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
2 | f1resrcmplf1dlem.3 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | ffnd 6669 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
4 | fnfvima 7182 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) | |
5 | 3, 4 | syl3an1 1163 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
6 | 1, 5 | syl3an2 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
7 | 6 | 3anidm12 1419 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
8 | 7 | ex 413 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐶 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))) |
9 | f1resrcmplf1dlem.2 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | |
10 | fnfvima 7182 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) | |
11 | 3, 10 | syl3an1 1163 | . . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
12 | 9, 11 | syl3an2 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
13 | 12 | 3anidm12 1419 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
14 | 13 | ex 413 | . 2 ⊢ (𝜑 → (𝑌 ∈ 𝐷 → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))) |
15 | f1resrcmplf1dlem.4 | . . . . 5 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅) | |
16 | disjne 4414 | . . . . 5 ⊢ ((((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅ ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) | |
17 | 15, 16 | syl3an1 1163 | . . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) |
18 | 17 | 3expib 1122 | . . 3 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))) |
19 | neneq 2949 | . . . 4 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) → ¬ (𝐹‘𝑋) = (𝐹‘𝑌)) | |
20 | 19 | pm2.21d 121 | . . 3 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)) |
21 | 18, 20 | syl6 35 | . 2 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))) |
22 | 8, 14, 21 | syl2and 608 | 1 ⊢ (𝜑 → ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 “ cima 5636 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 |
This theorem is referenced by: f1resrcmplf1d 33682 |
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