Theorem fconst4m 5648

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 5647	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))))
2		cnvimass 4910	. . . . . 6 ⊢ (◡𝐹 “ {𝐵}) ⊆ dom 𝐹
3		fndm 5230	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 3152	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) ⊆ 𝐴)
5	4	biantrurd 303	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐴 ⊆ (◡𝐹 “ {𝐵}) ↔ ((◡𝐹 “ {𝐵}) ⊆ 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))))
6		eqss 3117	. . . 4 ⊢ ((◡𝐹 “ {𝐵}) = 𝐴 ↔ ((◡𝐹 “ {𝐵}) ⊆ 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
7	5, 6	syl6bbr 197	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐴 ⊆ (◡𝐹 “ {𝐵}) ↔ (◡𝐹 “ {𝐵}) = 𝐴))
8	7	pm5.32i 450	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})) ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴))
9	1, 8	syl6bb 195	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481   ⊆ wss 3076  {csn 3532  ^◡ccnv 4546  dom cdm 4547   “ cima 4550   Fn wfn 5126  ⟶wf 5127

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139

This theorem is referenced by: (None)