Theorem fconst4 7169

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 7168	. 2 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
2		cnvimass 6047	. . . . . 6 ⊢ (◡𝐹 “ {𝐵}) ⊆ dom 𝐹
3		fndm 6601	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 3964	. . . . 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 3937	. . . 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 1542   ⊆ wss 3889  {csn 4567  ^◡ccnv 5630  dom cdm 5631   “ cima 5634   Fn wfn 6493  ⟶wf 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506

This theorem is referenced by:  lkr0f  39540