Theorem elrnmptg 5952

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 5948	. . 3 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
3	2	eleq2i 2861	. 2 ⊢ (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
4		r19.29 3134	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵))
5		eleq1 2857	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐶 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
6	5	biimparc 484	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ 𝑉)
7	6	elexd 3484	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ V)
8	7	rexlimivw 3168	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ V)
9	4, 8	syl 18	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) → 𝐶 ∈ V)
10	9	ex 417	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 𝐶 = 𝐵 → 𝐶 ∈ V))
11		eqeq1 2773	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐵 ↔ 𝐶 = 𝐵))
12	11	rexbidv 3195	. . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
13	12	elab3g 3651	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 → 𝐶 ∈ V) → (𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
14	10, 13	syl 18	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
15	3, 14	bitrid 286	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  Vcvv 3461   ↦ cmpt 5194  ran crn 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5112  df-opab 5176  df-mpt 5195  df-cnv 5670  df-dm 5672  df-rn 5673

This theorem is referenced by:  elrnmpti  5953  iunrnmptss  32851