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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1resrcmplf1dlem | Structured version Visualization version GIF version |
Description: Lemma for f1resrcmplf1d 32360. (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 6515 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
4 | fnfvima 6995 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) | |
5 | 3, 4 | syl3an1 1159 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
6 | 1, 5 | syl3an2 1160 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
7 | 6 | 3anidm12 1415 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) |
8 | 7 | ex 415 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐶 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))) |
9 | f1resrcmplf1dlem.2 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) | |
10 | fnfvima 6995 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) | |
11 | 3, 10 | syl3an1 1159 | . . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
12 | 9, 11 | syl3an2 1160 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
13 | 12 | 3anidm12 1415 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) |
14 | 13 | ex 415 | . 2 ⊢ (𝜑 → (𝑌 ∈ 𝐷 → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))) |
15 | f1resrcmplf1dlem.4 | . . . . 5 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅) | |
16 | disjne 4404 | . . . . 5 ⊢ ((((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅ ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) | |
17 | 15, 16 | syl3an1 1159 | . . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) |
18 | 17 | 3expib 1118 | . . 3 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))) |
19 | neneq 3022 | . . . 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 609 | 1 ⊢ (𝜑 → ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 “ cima 5558 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 |
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-sbc 3773 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-uni 4839 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-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 |
This theorem is referenced by: f1resrcmplf1d 32360 |
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