Theorem fvelrnb 5564

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 2461	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵))
2		19.41v 1902	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴))
3		simpl 109	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) → 𝑥 ∈ 𝐴)
4	3	anim1i 340	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝐹 Fn 𝐴))
5	4	ancomd 267	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴))
6		funfvex 5533	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
7	6	funfni 5317	. . . . . . . 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 110	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) → (𝐹‘𝑥) = 𝐵)
10	9	eleq1d 2246	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) → ((𝐹‘𝑥) ∈ V ↔ 𝐵 ∈ V))
11	10	adantr 276	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ((𝐹‘𝑥) ∈ V ↔ 𝐵 ∈ V))
12	8, 11	mpbid 147	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
13	12	exlimiv 1598	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
14	2, 13	sylbir 135	. . . 4 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
15	1, 14	sylanb 284	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
16	15	expcom 116	. 2 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → 𝐵 ∈ V))
17		fnrnfv 5563	. . . 4 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
18	17	eleq2d 2247	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}))
19		eqeq1 2184	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹‘𝑥) ↔ 𝐵 = (𝐹‘𝑥)))
20		eqcom 2179	. . . . . 6 ⊢ (𝐵 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝐵)
21	19, 20	bitrdi 196	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝐵))
22	21	rexbidv 2478	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
23	22	elab3g 2889	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
24	18, 23	sylan9bbr 463	. 2 ⊢ (((∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → 𝐵 ∈ V) ∧ 𝐹 Fn 𝐴) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
25	16, 24	mpancom 422	1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  {cab 2163  ∃wrex 2456  Vcvv 2738  ran crn 4628   Fn wfn 5212  ‘cfv 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225

This theorem is referenced by:  foelcdmi  5569  chfnrn  5628  rexrn  5654  ralrn  5655  elrnrexdmb  5657  ffnfv  5675  fconstfvm  5735  elunirn  5767  isoini  5819  canth  5829  reldm  6187  ordiso2  7034  eldju  7067  ctssdc  7112  uzn0  9543  frec2uzrand  10405  frecuzrdgtcl  10412  frecuzrdgfunlem  10419  uzin2  10996  reeff1o  14197