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| Description: Lemma for f1resrcmplf1d 35101. (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 6737 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴) | 
| 4 | fnfvima 7253 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) | |
| 5 | 3, 4 | syl3an1 1164 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) | 
| 6 | 1, 5 | syl3an2 1165 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)) | 
| 7 | 6 | 3anidm12 1421 | . . 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 1164 | . . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) | 
| 12 | 9, 11 | syl3an2 1165 | . . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) | 
| 13 | 12 | 3anidm12 1421 | . . 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 4455 | . . . . 5 ⊢ ((((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅ ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) | |
| 17 | 15, 16 | syl3an1 1164 | . . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)) | 
| 18 | 17 | 3expib 1123 | . . 3 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))) | 
| 19 | neneq 2946 | . . . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 | 
| This theorem is referenced by: f1resrcmplf1d 35101 | 
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