Theorem elrnmptg 5831

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 5827	. . 3 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
3	2	eleq2i 2904	. 2 ⊢ (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
4		r19.29 3254	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵))
5		eleq1 2900	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐶 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
6	5	biimparc 482	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ 𝑉)
7	6	elexd 3514	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ V)
8	7	rexlimivw 3282	. . . . 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 415	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 𝐶 = 𝐵 → 𝐶 ∈ V))
11		eqeq1 2825	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐵 ↔ 𝐶 = 𝐵))
12	11	rexbidv 3297	. . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
13	12	elab3g 3673	. . 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 285	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ↦ cmpt 5146  ran crn 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-mpt 5147  df-cnv 5563  df-dm 5565  df-rn 5566

This theorem is referenced by:  elrnmpti  5832  iunrnmptss  30317