Theorem fconst4 7160

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

Assertion

Ref	Expression
fconst4	⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴))

Proof of Theorem fconst4

Step	Hyp	Ref	Expression
1		fconst3 7159	. 2 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
2		cnvimass 6041	. . . . . 6 ⊢ (◡𝐹 “ {𝐵}) ⊆ dom 𝐹
3		fndm 6595	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 3976	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) ⊆ 𝐴)
5	4	biantrurd 532	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐴 ⊆ (◡𝐹 “ {𝐵}) ↔ ((◡𝐹 “ {𝐵}) ⊆ 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))))
6		eqss 3949	. . . 4 ⊢ ((◡𝐹 “ {𝐵}) = 𝐴 ↔ ((◡𝐹 “ {𝐵}) ⊆ 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
7	5, 6	bitr4di 289	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐴 ⊆ (◡𝐹 “ {𝐵}) ↔ (◡𝐹 “ {𝐵}) = 𝐴))
8	7	pm5.32i 574	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})) ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴))
9	1, 8	bitri 275	1 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ⊆ wss 3901  {csn 4580  ^◡ccnv 5623  dom cdm 5624   “ cima 5627   Fn wfn 6487  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  lkr0f  39350