Theorem fconst5 3843

Description: Two ways to express that a function is constant.

Assertion

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

Proof of Theorem fconst5

Step	Hyp	Ref	Expression
1		rnxp 3468	. . . . 5 ⊢ (A ≠ ∅ → ran ( A × {B}) = {B})
2	1	eqeq2d 1484	. . . 4 ⊢ (A ≠ ∅ → (ran F = ran ( A × {B}) ↔ ran F = {B}))
3		rneq 3335	. . . 4 ⊢ (F = (A × {B}) → ran F = ran ( A × {B}))
4	2, 3	syl5bi 208	. . 3 ⊢ (A ≠ ∅ → (F = (A × {B}) → ran F = {B}))
5	4	adantl 388	. 2 ⊢ ((F Fn A ⋀ A ≠ ∅) → (F = (A × {B}) → ran F = {B}))
6		fconst2g 3840	. . . . . 6 ⊢ (B ∈ V → (F:A–→{B} ↔ F = (A × {B})))
7		df-fo 3192	. . . . . . 7 ⊢ (F:A–onto→{B} ↔ (F Fn A ⋀ ran F = {B}))
8		fof 3667	. . . . . . 7 ⊢ (F:A–onto→{B} → F:A–→{B})
9	7, 8	sylbir 201	. . . . . 6 ⊢ ((F Fn A ⋀ ran F = {B}) → F:A–→{B})
10	6, 9	syl5bi 208	. . . . 5 ⊢ (B ∈ V → ((F Fn A ⋀ ran F = {B}) → F = (A × {B})))
11	10	exp3a 375	. . . 4 ⊢ (B ∈ V → (F Fn A → (ran F = {B} → F = (A × {B}))))
12	11	adantrd 391	. . 3 ⊢ (B ∈ V → ((F Fn A ⋀ A ≠ ∅) → (ran F = {B} → F = (A × {B}))))
13		snprc 2440	. . . . . 6 ⊢ (¬ B ∈ V ↔ {B} = ∅)
14		relrn0 3352	. . . . . . . . . 10 ⊢ (Rel F → (F = ∅ ↔ ran F = ∅))
15	14	biimprd 154	. . . . . . . . 9 ⊢ (Rel F → (ran F = ∅ → F = ∅))
16	15	adantl 388	. . . . . . . 8 ⊢ (({B} = ∅ ⋀ Rel F) → (ran F = ∅ → F = ∅))
17		eqeq2 1482	. . . . . . . . 9 ⊢ ({B} = ∅ → (ran F = {B} ↔ ran F = ∅))
18	17	adantr 389	. . . . . . . 8 ⊢ (({B} = ∅ ⋀ Rel F) → (ran F = {B} ↔ ran F = ∅))
19		xpeq2 3197	. . . . . . . . . . 11 ⊢ ({B} = ∅ → (A × {B}) = (A × ∅))
20		xp0 3461	. . . . . . . . . . 11 ⊢ (A × ∅) = ∅
21	19, 20	syl6eq 1521	. . . . . . . . . 10 ⊢ ({B} = ∅ → (A × {B}) = ∅)
22	21	eqeq2d 1484	. . . . . . . . 9 ⊢ ({B} = ∅ → (F = (A × {B}) ↔ F = ∅))
23	22	adantr 389	. . . . . . . 8 ⊢ (({B} = ∅ ⋀ Rel F) → (F = (A × {B}) ↔ F = ∅))
24	16, 18, 23	3imtr4d 542	. . . . . . 7 ⊢ (({B} = ∅ ⋀ Rel F) → (ran F = {B} → F = (A × {B})))
25	24	ex 373	. . . . . 6 ⊢ ({B} = ∅ → (Rel F → (ran F = {B} → F = (A × {B}))))
26	13, 25	sylbi 199	. . . . 5 ⊢ (¬ B ∈ V → (Rel F → (ran F = {B} → F = (A × {B}))))
27		fnrel 3582	. . . . 5 ⊢ (F Fn A → Rel F)
28	26, 27	syl5 21	. . . 4 ⊢ (¬ B ∈ V → (F Fn A → (ran F = {B} → F = (A × {B}))))
29	28	adantrd 391	. . 3 ⊢ (¬ B ∈ V → ((F Fn A ⋀ A ≠ ∅) → (ran F = {B} → F = (A × {B}))))
30	12, 29	pm2.61i 126	. 2 ⊢ ((F Fn A ⋀ A ≠ ∅) → (ran F = {B} → F = (A × {B})))
31	5, 30	impbid 515	1 ⊢ ((F Fn A ⋀ A ≠ ∅) → (F = (A × {B}) ↔ ran F = {B}))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 955   ∈ wcel 957   ≠ wne 1583  Vcvv 1808  ∅c0 2277  {csn 2406   × cxp 3164  ran crn 3167  Rel wrel 3171   Fn wfn 3173  –→wf 3174  –onto→wfo 3176

This theorem is referenced by:  nvo00 8384

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fo 3192  df-fv 3194

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