Theorem elimampt 6014

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 5651	. . 3 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
2	1	eleq2i 2820	. 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 5946	. . . . . . 7 ⊢ (𝐹 ↾ 𝐷) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷)
6		resmpt 6008	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵))
7	5, 6	eqtrid 2776	. . . . . 6 ⊢ (𝐷 ⊆ 𝐴 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵))
8	7	rneqd 5902	. . . . 5 ⊢ (𝐷 ⊆ 𝐴 → ran (𝐹 ↾ 𝐷) = ran (𝑥 ∈ 𝐷 ↦ 𝐵))
9	8	eleq2d 2814	. . . 4 ⊢ (𝐷 ⊆ 𝐴 → (𝐶 ∈ ran (𝐹 ↾ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ 𝐷 ↦ 𝐵)))
10	3, 9	syl 17	. . 3 ⊢ (𝜑 → (𝐶 ∈ ran (𝐹 ↾ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ 𝐷 ↦ 𝐵)))
11		elimampt.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
12		eqid 2729	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
13	12	elrnmpt 5922	. . . 4 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ ran (𝑥 ∈ 𝐷 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))
14	11, 13	syl 17	. . 3 ⊢ (𝜑 → (𝐶 ∈ ran (𝑥 ∈ 𝐷 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))
15	10, 14	bitrd 279	. 2 ⊢ (𝜑 → (𝐶 ∈ ran (𝐹 ↾ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))
16	2, 15	bitrid 283	1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3914   ↦ cmpt 5188  ran crn 5639   ↾ cres 5640   “ cima 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-mpt 5189  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651

This theorem is referenced by:  reprpmtf1o  34617  ellcsrspsn  35628