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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 5532 | . . 3 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷) | |
2 | 1 | eleq2i 2881 | . 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 5814 | . . . . . . 7 ⊢ (𝐹 ↾ 𝐷) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷) |
6 | resmpt 5872 | . . . . . . 7 ⊢ (𝐷 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵)) | |
7 | 5, 6 | syl5eq 2845 | . . . . . 6 ⊢ (𝐷 ⊆ 𝐴 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵)) |
8 | 7 | rneqd 5772 | . . . . 5 ⊢ (𝐷 ⊆ 𝐴 → ran (𝐹 ↾ 𝐷) = ran (𝑥 ∈ 𝐷 ↦ 𝐵)) |
9 | 8 | eleq2d 2875 | . . . 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 2798 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
13 | 12 | elrnmpt 5792 | . . . 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 282 | . 2 ⊢ (𝜑 → (𝐶 ∈ ran (𝐹 ↾ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵)) |
16 | 2, 15 | syl5bb 286 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ⊆ wss 3881 ↦ cmpt 5110 ran crn 5520 ↾ cres 5521 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: reprpmtf1o 32007 |
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