Theorem elrnmpt2d 4987

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 4981	. . 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 1397   ∈ wcel 2202  ∃wrex 2511   ↦ cmpt 4150  ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-mpt 4152  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by: (None)