Theorem elrnmptg 5029

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 5025	. . 3 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
3	2	eleq2i 2305	. 2 ⊢ (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
4		r19.29 2688	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵))
5		eleq1 2301	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐶 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
6	5	biimparc 299	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ 𝑉)
7		elex 2833	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
8	6, 7	syl 14	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ V)
9	8	rexlimivw 2664	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ V)
10	4, 9	syl 14	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) → 𝐶 ∈ V)
11	10	ex 115	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 𝐶 = 𝐵 → 𝐶 ∈ V))
12		eqeq1 2245	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐵 ↔ 𝐶 = 𝐵))
13	12	rexbidv 2551	. . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
14	13	elab3g 2977	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 → 𝐶 ∈ V) → (𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
15	11, 14	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
16	3, 15	bitrid 192	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ↦ cmpt 4187  ran crn 4770

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-mpt 4189  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  elrnmpti  5030  fliftel  5989  2sqlem1  16147