Theorem fconst4m 5779

Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)

Assertion

Ref	Expression
fconst4m	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴)))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem fconst4m

Step	Hyp	Ref	Expression
1		fconst3m 5778	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))))
2		cnvimass 5029	. . . . . 6 ⊢ (◡𝐹 “ {𝐵}) ⊆ dom 𝐹
3		fndm 5354	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 3230	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) ⊆ 𝐴)
5	4	biantrurd 305	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐴 ⊆ (◡𝐹 “ {𝐵}) ↔ ((◡𝐹 “ {𝐵}) ⊆ 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))))
6		eqss 3195	. . . 4 ⊢ ((◡𝐹 “ {𝐵}) = 𝐴 ↔ ((◡𝐹 “ {𝐵}) ⊆ 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
7	5, 6	bitr4di 198	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐴 ⊆ (◡𝐹 “ {𝐵}) ↔ (◡𝐹 “ {𝐵}) = 𝐴))
8	7	pm5.32i 454	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})) ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴))
9	1, 8	bitrdi 196	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164   ⊆ wss 3154  {csn 3619  ^◡ccnv 4659  dom cdm 4660   “ cima 4663   Fn wfn 5250  ⟶wf 5251

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263

This theorem is referenced by: (None)