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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1resrcmplf1dlem | Structured version Visualization version GIF version |
Description: Lemma for f1resrcmplf1d 35080. (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 6738 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
4 | fnfvima 7253 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) | |
5 | 3, 4 | syl3an1 1162 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
6 | 1, 5 | syl3an2 1163 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
7 | 6 | 3anidm12 1418 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
8 | 7 | ex 412 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐶 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))) |
9 | f1resrcmplf1dlem.2 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | |
10 | fnfvima 7253 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) | |
11 | 3, 10 | syl3an1 1162 | . . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
12 | 9, 11 | syl3an2 1163 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
13 | 12 | 3anidm12 1418 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
14 | 13 | ex 412 | . 2 ⊢ (𝜑 → (𝑌 ∈ 𝐷 → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))) |
15 | f1resrcmplf1dlem.4 | . . . . 5 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅) | |
16 | disjne 4461 | . . . . 5 ⊢ ((((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅ ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) | |
17 | 15, 16 | syl3an1 1162 | . . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) |
18 | 17 | 3expib 1121 | . . 3 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))) |
19 | neneq 2944 | . . . 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 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 |
This theorem is referenced by: f1resrcmplf1d 35080 |
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