Theorem fconst5 7225

Description: Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)

Assertion

Ref	Expression
fconst5	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))

Proof of Theorem fconst5

Step	Hyp	Ref	Expression
1		rneq 5944	. . . 4 ⊢ (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = ran (𝐴 × {𝐵}))
2		rnxp 6183	. . . . 5 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
3	2	eqeq2d 2737	. . . 4 ⊢ (𝐴 ≠ ∅ → (ran 𝐹 = ran (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
4	1, 3	imbitrid 243	. . 3 ⊢ (𝐴 ≠ ∅ → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
5	4	adantl 480	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) → ran 𝐹 = {𝐵}))
6		df-fo 6562	. . . . . . 7 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
7		fof 6817	. . . . . . 7 ⊢ (𝐹:𝐴–onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
8	6, 7	sylbir 234	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹:𝐴⟶{𝐵})
9		fconst2g 7222	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
10	8, 9	imbitrid 243	. . . . 5 ⊢ (𝐵 ∈ V → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹 = (𝐴 × {𝐵})))
11	10	expd 414	. . . 4 ⊢ (𝐵 ∈ V → (𝐹 Fn 𝐴 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
12	11	adantrd 490	. . 3 ⊢ (𝐵 ∈ V → ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
13		fnrel 6664	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
14		snprc 4726	. . . . . 6 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
15		relrn0 5978	. . . . . . . . . 10 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ ran 𝐹 = ∅))
16	15	biimprd 247	. . . . . . . . 9 ⊢ (Rel 𝐹 → (ran 𝐹 = ∅ → 𝐹 = ∅))
17	16	adantl 480	. . . . . . . 8 ⊢ (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = ∅ → 𝐹 = ∅))
18		eqeq2 2738	. . . . . . . . 9 ⊢ ({𝐵} = ∅ → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
19	18	adantr 479	. . . . . . . 8 ⊢ (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} ↔ ran 𝐹 = ∅))
20		xpeq2 5705	. . . . . . . . . . 11 ⊢ ({𝐵} = ∅ → (𝐴 × {𝐵}) = (𝐴 × ∅))
21		xp0 6171	. . . . . . . . . . 11 ⊢ (𝐴 × ∅) = ∅
22	20, 21	eqtrdi 2782	. . . . . . . . . 10 ⊢ ({𝐵} = ∅ → (𝐴 × {𝐵}) = ∅)
23	22	eqeq2d 2737	. . . . . . . . 9 ⊢ ({𝐵} = ∅ → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
24	23	adantr 479	. . . . . . . 8 ⊢ (({𝐵} = ∅ ∧ Rel 𝐹) → (𝐹 = (𝐴 × {𝐵}) ↔ 𝐹 = ∅))
25	17, 19, 24	3imtr4d 293	. . . . . . 7 ⊢ (({𝐵} = ∅ ∧ Rel 𝐹) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
26	25	ex 411	. . . . . 6 ⊢ ({𝐵} = ∅ → (Rel 𝐹 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
27	14, 26	sylbi 216	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (Rel 𝐹 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
28	13, 27	syl5 34	. . . 4 ⊢ (¬ 𝐵 ∈ V → (𝐹 Fn 𝐴 → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
29	28	adantrd 490	. . 3 ⊢ (¬ 𝐵 ∈ V → ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵}))))
30	12, 29	pm2.61i 182	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (ran 𝐹 = {𝐵} → 𝐹 = (𝐴 × {𝐵})))
31	5, 30	impbid 211	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  Vcvv 3462  ∅c0 4325  {csn 4633   × cxp 5682  ran crn 5685  Rel wrel 5689   Fn wfn 6551  ⟶wf 6552  –onto→wfo 6554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-fo 6562  df-fv 6564

This theorem is referenced by:  imadrhmcl  20778  nvo00  30697  zar0ring  33695  esumnul  33883  esum0  33884  volsupnfl  37368