Theorem elrnmpt1sf 39889

Description: Elementhood in an image set. Same as elrnmpt1s 5509, 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 2771	. . 3 ⊢ 𝐶 = 𝐶
2	1	nfth 1875	. . . 4 ⊢ Ⅎ𝑥 𝐶 = 𝐶
3		elrnmpt1sf.3	. . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
4	3	eqeq2d 2781	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶))
5	2, 4	rspce 3455	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
6	1, 5	mpan2 671	. 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
7		elrnmpt1sf.1	. . . 4 ⊢ Ⅎ𝑥𝐶
8		elrnmpt1sf.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
9	7, 8	elrnmptf 39880	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
10	9	biimparc 465	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)
11	6, 10	sylan 569	1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  Ⅎwnfc 2900  ∃wrex 3062   ↦ cmpt 4863  ran crn 5250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-mpt 4864  df-cnv 5257  df-dm 5259  df-rn 5260

This theorem is referenced by:  sge0f1o  41109