Theorem fvelrnb 5611

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 2481	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵))
2		19.41v 1917	. . . . 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 5578	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
7	6	funfni 5361	. . . . . . . 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 2265	. . . . . . . 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 1612	. . . . 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 5610	. . . 4 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
18	17	eleq2d 2266	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}))
19		eqeq1 2203	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹‘𝑥) ↔ 𝐵 = (𝐹‘𝑥)))
20		eqcom 2198	. . . . . 6 ⊢ (𝐵 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝐵)
21	19, 20	bitrdi 196	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝐵))
22	21	rexbidv 2498	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
23	22	elab3g 2915	. . 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 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182  ∃wrex 2476  Vcvv 2763  ran crn 4665   Fn wfn 5254  ‘cfv 5259

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267

This theorem is referenced by:  foelcdmi  5616  chfnrn  5676  rexrn  5702  ralrn  5703  elrnrexdmb  5705  ffnfv  5723  fconstfvm  5783  elunirn  5816  isoini  5868  canth  5878  reldm  6253  ordiso2  7110  eldju  7143  ctssdc  7188  uzn0  9634  frec2uzrand  10514  frecuzrdgtcl  10521  frecuzrdgfunlem  10528  uzin2  11169  imasmnd2  13154  imasgrp2  13316  imasrng  13588  imasring  13696  reeff1o  15093