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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincmoALT | Structured version Visualization version GIF version |
Description: Alternate proof for thincmo 48147. (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 48139 | . . . 4 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))) |
5 | 4 | simprbi 495 | . . 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 7425 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) | |
10 | 9 | eleq2d 2811 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑋𝐻𝑌))) |
11 | 10 | mobidv 2537 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
12 | 11 | rspc2gv 3611 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
13 | 7, 8, 12 | syl2anc 582 | . 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 394 = wceq 1533 ∈ wcel 2098 ∃*wmo 2526 ∀wral 3051 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 Hom chom 17243 Catccat 17643 ThinCatcthinc 48137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-nul 5301 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2528 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-iota 6495 df-fv 6551 df-ov 7419 df-thinc 48138 |
This theorem is referenced by: (None) |
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