Theorem elimampt 30382

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 5567	. . 3 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
2	1	eleq2i 2904	. 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 5848	. . . . . . 7 ⊢ (𝐹 ↾ 𝐷) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷)
6		resmpt 5904	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵))
7	5, 6	syl5eq 2868	. . . . . 6 ⊢ (𝐷 ⊆ 𝐴 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵))
8	7	rneqd 5807	. . . . 5 ⊢ (𝐷 ⊆ 𝐴 → ran (𝐹 ↾ 𝐷) = ran (𝑥 ∈ 𝐷 ↦ 𝐵))
9	8	eleq2d 2898	. . . 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 2821	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
13	12	elrnmpt 5827	. . . 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 281	. 2 ⊢ (𝜑 → (𝐶 ∈ ran (𝐹 ↾ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))
16	2, 15	syl5bb 285	1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∃wrex 3139   ⊆ wss 3935   ↦ cmpt 5145  ran crn 5555   ↾ cres 5556   “ cima 5557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-mpt 5146  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567

This theorem is referenced by:  reprpmtf1o  31897