Theorem elrnmpt1sf 46021

Description: Elementhood in an image set. Same as elrnmpt1s 5943, 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 2760	. . 3 ⊢ 𝐶 = 𝐶
2		elrnmpt1sf.1	. . . . 5 ⊢ Ⅎ𝑥𝐶
3	2, 2	nfeq 2935	. . . 4 ⊢ Ⅎ𝑥 𝐶 = 𝐶
4		elrnmpt1sf.3	. . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
5	4	eqeq2d 2771	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶))
6	3, 5	rspce 3565	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
7	1, 6	mpan2 704	. 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
8		elrnmpt1sf.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
9	2, 8	elrnmptf 46013	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
10	9	biimparc 485	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)
11	7, 10	sylan 592	1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2907  ∃wrex 3086   ↦ cmpt 5186  ran crn 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-mpt 5187  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  sge0f1o  47210