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Mirrors > Home > MPE Home > Th. List > Mathboxes > elimampt | Structured version Visualization version GIF version |
Description: Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.) |
Ref | Expression |
---|---|
elimampt.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
elimampt.c | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
elimampt.d | ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
Ref | Expression |
---|---|
elimampt | ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5568 | . . 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 5849 | . . . . . . 7 ⊢ (𝐹 ↾ 𝐷) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷) |
6 | resmpt 5905 | . . . . . . 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 5808 | . . . . 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 5828 | . . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ⊆ wss 3936 ↦ cmpt 5146 ran crn 5556 ↾ cres 5557 “ cima 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: reprpmtf1o 31897 |
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