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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1resrcmplf1dlem | Structured version Visualization version GIF version |
Description: Lemma for f1resrcmplf1d 32744. (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 6535 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
4 | fnfvima 7038 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) | |
5 | 3, 4 | syl3an1 1165 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
6 | 1, 5 | syl3an2 1166 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
7 | 6 | 3anidm12 1421 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
8 | 7 | ex 416 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐶 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))) |
9 | f1resrcmplf1dlem.2 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | |
10 | fnfvima 7038 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) | |
11 | 3, 10 | syl3an1 1165 | . . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
12 | 9, 11 | syl3an2 1166 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
13 | 12 | 3anidm12 1421 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
14 | 13 | ex 416 | . 2 ⊢ (𝜑 → (𝑌 ∈ 𝐷 → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))) |
15 | f1resrcmplf1dlem.4 | . . . . 5 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅) | |
16 | disjne 4359 | . . . . 5 ⊢ ((((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅ ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) | |
17 | 15, 16 | syl3an1 1165 | . . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) |
18 | 17 | 3expib 1124 | . . 3 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))) |
19 | neneq 2941 | . . . 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 611 | 1 ⊢ (𝜑 → ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∩ cin 3856 ⊆ wss 3857 ∅c0 4227 “ cima 5543 Fn wfn 6364 ⟶wf 6365 ‘cfv 6369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 |
This theorem is referenced by: f1resrcmplf1d 32744 |
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