Theorem elimampt 6045

Description: Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.)

Hypotheses

Ref	Expression
elimampt.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
elimampt.c	⊢ (𝜑 → 𝐶 ∈ 𝑊)
elimampt.d	⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Assertion

Ref	Expression
elimampt	⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷

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

Proof of Theorem elimampt

Step	Hyp	Ref	Expression
1		df-ima 5674	. . 3 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
2	1	eleq2i 2853	. 2 ⊢ (𝐶 ∈ (𝐹 “ 𝐷) ↔ 𝐶 ∈ ran (𝐹 ↾ 𝐷))
3		elimampt.d	. . . 4 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
4		elimampt.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	reseq1i 5974	. . . . . . 7 ⊢ (𝐹 ↾ 𝐷) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷)
6		resmpt 6039	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵))
7	5, 6	eqtrid 2808	. . . . . 6 ⊢ (𝐷 ⊆ 𝐴 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵))
8	7	rneqd 5928	. . . . 5 ⊢ (𝐷 ⊆ 𝐴 → ran (𝐹 ↾ 𝐷) = ran (𝑥 ∈ 𝐷 ↦ 𝐵))
9	8	eleq2d 2847	. . . 4 ⊢ (𝐷 ⊆ 𝐴 → (𝐶 ∈ ran (𝐹 ↾ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ 𝐷 ↦ 𝐵)))
10	3, 9	syl 18	. . 3 ⊢ (𝜑 → (𝐶 ∈ ran (𝐹 ↾ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ 𝐷 ↦ 𝐵)))
11		elimampt.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
12		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
13	12	elrnmpt 5948	. . . 4 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ ran (𝑥 ∈ 𝐷 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))
14	11, 13	syl 18	. . 3 ⊢ (𝜑 → (𝐶 ∈ ran (𝑥 ∈ 𝐷 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))
15	10, 14	bitrd 282	. 2 ⊢ (𝜑 → (𝐶 ∈ ran (𝐹 ↾ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))
16	2, 15	bitrid 286	1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568   ∈ wcel 2141  ∃wrex 3087   ⊆ wss 3904   ↦ cmpt 5191  ran crn 5662   ↾ cres 5663   “ cima 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-mpt 5192  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674

This theorem is referenced by:  reprpmtf1o  34979  ellcsrspsn  36087