Theorem fconst3m 5903

Description: Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)

Assertion

Ref	Expression
fconst3m	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fconst3m

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconstfvm 5902	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵)))
2		fnfun 5453	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3		fndm 5455	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4		eqimss2 3293	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹)
5	3, 4	syl 14	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐴 ⊆ dom 𝐹)
6		funconstss 5796	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
7	2, 5, 6	syl2anc 411	. . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
8	7	pm5.32i 454	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
9	1, 8	bitrdi 196	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520   ⊆ wss 3211  {csn 3689  ^◡ccnv 4748  dom cdm 4749   “ cima 4752  Fun wfun 5346   Fn wfn 5347  ⟶wf 5348  ‘cfv 5352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360

This theorem is referenced by:  fconst4m  5904