Theorem fconst4 7216

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 7215	. 2 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
2		cnvimass 6081	. . . . . 6 ⊢ (◡𝐹 “ {𝐵}) ⊆ dom 𝐹
3		fndm 6653	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 4035	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) ⊆ 𝐴)
5	4	biantrurd 534	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐴 ⊆ (◡𝐹 “ {𝐵}) ↔ ((◡𝐹 “ {𝐵}) ⊆ 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵}))))
6		eqss 3998	. . . 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 576	. 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 205   ∧ wa 397   = wceq 1542   ⊆ wss 3949  {csn 4629  ^◡ccnv 5676  dom cdm 5677   “ cima 5680   Fn wfn 6539  ⟶wf 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552

This theorem is referenced by:  lkr0f  37964