Theorem fconst3 7216

Description: Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)

Assertion

Ref	Expression
fconst3	⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))

Proof of Theorem fconst3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconstfv 7215	. 2 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
2		fnfun 6649	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3		fndm 6652	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4		eqimss2 4025	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹)
5	3, 4	syl 17	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐴 ⊆ dom 𝐹)
6		funconstss 7057	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
7	2, 5, 6	syl2anc 584	. . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
8	7	pm5.32i 574	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
9	1, 8	bitri 275	1 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539  ∀wral 3050   ⊆ wss 3933  {csn 4608  ^◡ccnv 5666  dom cdm 5667   “ cima 5670  Fun wfun 6536   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550

This theorem is referenced by:  fconst4  7217  dnsconst  23351