Theorem fconst3m 5857

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 5856	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝐵)))
2		fnfun 5417	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3		fndm 5419	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4		eqimss2 3279	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹)
5	3, 4	syl 14	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐴 ⊆ dom 𝐹)
6		funconstss 5752	. . . 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 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  {csn 3666  ^◡ccnv 4717  dom cdm 4718   “ cima 4721  Fun wfun 5311   Fn wfn 5312  ⟶wf 5313  ‘cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fo 5323  df-fv 5325

This theorem is referenced by:  fconst4m  5858