Theorem dftermc3 50682

Description: Alternate definition of TermCat. See also df-termc 50624, dftermc2 50671. (Contributed by Zhi Wang, 20-Oct-2025.)

Assertion

Ref	Expression
dftermc3	⊢ TermCat = {𝑐 ∣ (Arrow‘𝑐) ≈ 1o}

Proof of Theorem dftermc3

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcarweu 50679	. . . 4 ⊢ (𝑐 ∈ TermCat → ∃!𝑎 𝑎 ∈ (Arrow‘𝑐))
2		arweutermc 50681	. . . 4 ⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝑐) → 𝑐 ∈ TermCat)
3	1, 2	impbii 212	. . 3 ⊢ (𝑐 ∈ TermCat ↔ ∃!𝑎 𝑎 ∈ (Arrow‘𝑐))
4		euen1b 9055	. . 3 ⊢ ((Arrow‘𝑐) ≈ 1o ↔ ∃!𝑎 𝑎 ∈ (Arrow‘𝑐))
5	3, 4	bitr4i 281	. 2 ⊢ (𝑐 ∈ TermCat ↔ (Arrow‘𝑐) ≈ 1o)
6	5	eqabi 2895	1 ⊢ TermCat = {𝑐 ∣ (Arrow‘𝑐) ≈ 1o}

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  ∃!weu 2593  {cab 2738   class class class wbr 5102  ‘cfv 6538  1_oc1o 8469   ≈ cen 8970  Arrowcarw 18197  TermCatctermc 50623

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-ot 4592  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-1st 8001  df-2nd 8002  df-1o 8476  df-en 8974  df-cat 17842  df-cid 17843  df-doma 18199  df-coda 18200  df-homa 18201  df-arw 18202  df-thinc 50569  df-termc 50624

This theorem is used by: (None)