Theorem elrnmpt1sf 44186

Description: Elementhood in an image set. Same as elrnmpt1s 5955, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
elrnmpt1sf.1	⊢ Ⅎ𝑥𝐶
elrnmpt1sf.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
elrnmpt1sf.3	⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)

Assertion

Ref	Expression
elrnmpt1sf	⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐷

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

Proof of Theorem elrnmpt1sf

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ 𝐶 = 𝐶
2		elrnmpt1sf.1	. . . . 5 ⊢ Ⅎ𝑥𝐶
3	2, 2	nfeq 2914	. . . 4 ⊢ Ⅎ𝑥 𝐶 = 𝐶
4		elrnmpt1sf.3	. . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
5	4	eqeq2d 2741	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶))
6	3, 5	rspce 3600	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
7	1, 6	mpan2 687	. 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
8		elrnmpt1sf.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
9	2, 8	elrnmptf 44178	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
10	9	biimparc 478	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)
11	7, 10	sylan 578	1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  Ⅎwnfc 2881  ∃wrex 3068   ↦ cmpt 5230  ran crn 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-mpt 5231  df-cnv 5683  df-dm 5685  df-rn 5686

This theorem is referenced by:  sge0f1o  45396