Theorem fconst3m 5639

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 5638	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵)))
2		fnfun 5220	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3		fndm 5222	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4		eqimss2 3152	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹)
5	3, 4	syl 14	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐴 ⊆ dom 𝐹)
6		funconstss 5538	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
7	2, 5, 6	syl2anc 408	. . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
8	7	pm5.32i 449	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
9	1, 8	syl6bb 195	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416   ⊆ wss 3071  {csn 3527  ^◡ccnv 4538  dom cdm 4539   “ cima 4542  Fun wfun 5117   Fn wfn 5118  ⟶wf 5119  ‘cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131

This theorem is referenced by:  fconst4m  5640