Theorem fvelrnb 5544

Description: A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)

Assertion

Ref	Expression
fvelrnb	⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fvelrnb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2454	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵))
2		19.41v 1895	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴))
3		simpl 108	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) → 𝑥 ∈ 𝐴)
4	3	anim1i 338	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝐹 Fn 𝐴))
5	4	ancomd 265	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴))
6		funfvex 5513	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
7	6	funfni 5298	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
8	5, 7	syl 14	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹‘𝑥) ∈ V)
9		simpr 109	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) → (𝐹‘𝑥) = 𝐵)
10	9	eleq1d 2239	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) → ((𝐹‘𝑥) ∈ V ↔ 𝐵 ∈ V))
11	10	adantr 274	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ((𝐹‘𝑥) ∈ V ↔ 𝐵 ∈ V))
12	8, 11	mpbid 146	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
13	12	exlimiv 1591	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
14	2, 13	sylbir 134	. . . 4 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
15	1, 14	sylanb 282	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
16	15	expcom 115	. 2 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → 𝐵 ∈ V))
17		fnrnfv 5543	. . . 4 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
18	17	eleq2d 2240	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}))
19		eqeq1 2177	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹‘𝑥) ↔ 𝐵 = (𝐹‘𝑥)))
20		eqcom 2172	. . . . . 6 ⊢ (𝐵 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝐵)
21	19, 20	bitrdi 195	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝐵))
22	21	rexbidv 2471	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
23	22	elab3g 2881	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
24	18, 23	sylan9bbr 460	. 2 ⊢ (((∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → 𝐵 ∈ V) ∧ 𝐹 Fn 𝐴) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
25	16, 24	mpancom 420	1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  ∃wrex 2449  Vcvv 2730  ran crn 4612   Fn wfn 5193  ‘cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  chfnrn  5607  rexrn  5633  ralrn  5634  elrnrexdmb  5636  ffnfv  5654  fconstfvm  5714  elunirn  5745  isoini  5797  canth  5807  reldm  6165  ordiso2  7012  eldju  7045  ctssdc  7090  uzn0  9502  frec2uzrand  10361  frecuzrdgtcl  10368  frecuzrdgfunlem  10375  uzin2  10951  reeff1o  13488