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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincmoALT | Structured version Visualization version GIF version |
Description: Alternate proof for thincmo 46310. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
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 |
---|---|
thincmoALT | ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thincmo.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
2 | thincmo.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
3 | thincmo.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | 2, 3 | isthinc 46302 | . . . 4 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))) |
5 | 4 | simprbi 497 | . . 3 ⊢ (𝐶 ∈ ThinCat → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) |
7 | thincmo.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | thincmo.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | oveq12 7284 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) | |
10 | 9 | eleq2d 2824 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑋𝐻𝑌))) |
11 | 10 | mobidv 2549 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
12 | 11 | rspc2gv 3569 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
13 | 7, 8, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
14 | 6, 13 | mpd 15 | 1 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∃*wmo 2538 ∀wral 3064 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Hom chom 16973 Catccat 17373 ThinCatcthinc 46300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 df-thinc 46301 |
This theorem is referenced by: (None) |
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