Theorem elrnmptg 5857

Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
rnmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
elrnmptg	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmptg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnmpt.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	rnmpt 5853	. . 3 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
3	2	eleq2i 2830	. 2 ⊢ (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
4		r19.29 3183	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵))
5		eleq1 2826	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐶 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
6	5	biimparc 479	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ 𝑉)
7	6	elexd 3442	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ V)
8	7	rexlimivw 3210	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ V)
9	4, 8	syl 17	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) → 𝐶 ∈ V)
10	9	ex 412	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 𝐶 = 𝐵 → 𝐶 ∈ V))
11		eqeq1 2742	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐵 ↔ 𝐶 = 𝐵))
12	11	rexbidv 3225	. . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
13	12	elab3g 3609	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 → 𝐶 ∈ V) → (𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
14	10, 13	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
15	3, 14	syl5bb 282	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ↦ cmpt 5153  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-mpt 5154  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  elrnmpti  5858  iunrnmptss  30806