Theorem elrnmpt1sf 45310

Description: Elementhood in an image set. Same as elrnmpt1s 5903, 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 2733	. . 3 ⊢ 𝐶 = 𝐶
2		elrnmpt1sf.1	. . . . 5 ⊢ Ⅎ𝑥𝐶
3	2, 2	nfeq 2909	. . . 4 ⊢ Ⅎ𝑥 𝐶 = 𝐶
4		elrnmpt1sf.3	. . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
5	4	eqeq2d 2744	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶))
6	3, 5	rspce 3562	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
7	1, 6	mpan2 691	. 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
8		elrnmpt1sf.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
9	2, 8	elrnmptf 45302	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
10	9	biimparc 479	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)
11	7, 10	sylan 580	1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2880  ∃wrex 3057   ↦ cmpt 5174  ran crn 5620

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-mpt 5175  df-cnv 5627  df-dm 5629  df-rn 5630

This theorem is referenced by:  sge0f1o  46504