Theorem thincmoALT 49919

Description: Alternate proof of thincmo 49918. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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)

Assertion

Ref	Expression
thincmoALT	⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))

Distinct variable groups:   𝐵,𝑓   𝐶,𝑓   𝑓,𝐻   𝑓,𝑋   𝑓,𝑌   𝜑,𝑓

Proof of Theorem thincmoALT

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

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 49909	. . . 4 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
5	4	simprbi 498	. . 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 7365	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
10	9	eleq2d 2825	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑋𝐻𝑌)))
11	10	mobidv 2553	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
12	11	rspc2gv 3570	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
13	7, 8, 12	syl2anc 590	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
14	6, 13	mpd 15	1 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541  ∀wral 3053  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  Hom chom 17222  Catccat 17621  ThinCatcthinc 49907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-thinc 49908

This theorem is referenced by: (None)