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Theorem thincmoALT 47929
Description: Alternate proof for thincmo 47928. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
thincmo.c (𝜑𝐶 ∈ ThinCat)
thincmo.x (𝜑𝑋𝐵)
thincmo.y (𝜑𝑌𝐵)
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.
StepHypRef Expression
1 thincmo.c . . 3 (𝜑𝐶 ∈ ThinCat)
2 thincmo.b . . . . 5 𝐵 = (Base‘𝐶)
3 thincmo.h . . . . 5 𝐻 = (Hom ‘𝐶)
42, 3isthinc 47920 . . . 4 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
54simprbi 496 . . 3 (𝐶 ∈ ThinCat → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
61, 5syl 17 . 2 (𝜑 → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
7 thincmo.x . . 3 (𝜑𝑋𝐵)
8 thincmo.y . . 3 (𝜑𝑌𝐵)
9 oveq12 7414 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
109eleq2d 2813 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑋𝐻𝑌)))
1110mobidv 2537 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
1211rspc2gv 3616 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
137, 8, 12syl2anc 583 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
146, 13mpd 15 1 (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  ∃*wmo 2526  wral 3055  cfv 6537  (class class class)co 7405  Basecbs 17153  Hom chom 17217  Catccat 17617  ThinCatcthinc 47918
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 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-thinc 47919
This theorem is referenced by: (None)
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