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Mirrors > Home > MPE Home > Th. List > istermoi | Structured version Visualization version GIF version |
Description: Implication of a class being a terminal object. (Contributed by AV, 18-Apr-2020.) |
Ref | Expression |
---|---|
isinitoi.b | ⊢ 𝐵 = (Base‘𝐶) |
isinitoi.h | ⊢ 𝐻 = (Hom ‘𝐶) |
isinitoi.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
Ref | Expression |
---|---|
istermoi | ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinitoi.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
2 | isinitoi.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
3 | isinitoi.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | 1, 2, 3 | termoval 17261 | . . . . 5 ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)}) |
5 | 4 | eleq2d 2901 | . . . 4 ⊢ (𝜑 → (𝑂 ∈ (TermO‘𝐶) ↔ 𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})) |
6 | elrabi 3678 | . . . 4 ⊢ (𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} → 𝑂 ∈ 𝐵) | |
7 | 5, 6 | syl6bi 255 | . . 3 ⊢ (𝜑 → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵)) |
8 | 7 | imp 409 | . 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ 𝐵) |
9 | 1 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat) |
10 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵) | |
11 | 2, 3, 9, 10 | istermo 17264 | . . . 4 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂))) |
12 | 11 | biimpd 231 | . . 3 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (TermO‘𝐶) → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂))) |
13 | 12 | impancom 454 | . 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂))) |
14 | 8, 13 | jcai 519 | 1 ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃!weu 2652 ∀wral 3141 {crab 3145 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 Hom chom 16579 Catccat 16938 TermOctermo 17252 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-termo 17255 |
This theorem is referenced by: termoid 17269 termoo 17271 termoeu1 17281 |
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