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| Mirrors > Home > MPE Home > Th. List > Mathboxes > thincmo | Structured version Visualization version GIF version | ||
| Description: There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| thincmo.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) | 
| thincmo.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| thincmo.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| thincmo.b | ⊢ 𝐵 = (Base‘𝐶) | 
| thincmo.h | ⊢ 𝐻 = (Hom ‘𝐶) | 
| Ref | Expression | 
|---|---|
| thincmo | ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | thincmo.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 2 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑋 ∈ 𝐵) | 
| 3 | thincmo.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 4 | 3 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑌 ∈ 𝐵) | 
| 5 | simprl 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 ∈ (𝑋𝐻𝑌)) | |
| 6 | simprr 772 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑔 ∈ (𝑋𝐻𝑌)) | |
| 7 | thincmo.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
| 8 | thincmo.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
| 9 | thincmo.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
| 10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝐶 ∈ ThinCat) | 
| 11 | 2, 4, 5, 6, 7, 8, 10 | thincmo2 49100 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 = 𝑔) | 
| 12 | 11 | ex 412 | . . 3 ⊢ (𝜑 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔)) | 
| 13 | 12 | alrimivv 1927 | . 2 ⊢ (𝜑 → ∀𝑓∀𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔)) | 
| 14 | eleq1w 2823 | . . 3 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑔 ∈ (𝑋𝐻𝑌))) | |
| 15 | 14 | mo4 2565 | . 2 ⊢ (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∀𝑓∀𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔)) | 
| 16 | 13, 15 | sylibr 234 | 1 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 ∃*wmo 2537 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Hom chom 17309 ThinCatcthinc 49091 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-thinc 49092 | 
| This theorem is referenced by: thincmod 49104 oppcthin 49112 subthinc 49117 functhinclem1 49118 functhinclem4 49121 thincfth 49126 thincciso2 49129 | 
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