Theorem elrnmpt2d 4921

Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypotheses

Ref	Expression
elrnmpt2d.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
elrnmpt2d.2	⊢ (𝜑 → 𝐶 ∈ ran 𝐹)

Assertion

Ref	Expression
elrnmpt2d	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem elrnmpt2d

Step	Hyp	Ref	Expression
1		elrnmpt2d.2	. 2 ⊢ (𝜑 → 𝐶 ∈ ran 𝐹)
2		elrnmpt2d.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	elrnmpt 4915	. . 3 ⊢ (𝐶 ∈ ran 𝐹 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
4	3	ibi 176	. 2 ⊢ (𝐶 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
5	1, 4	syl 14	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  ∃wrex 2476   ↦ cmpt 4094  ran crn 4664

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-mpt 4096  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by: (None)