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| Mirrors > Home > MPE Home > Th. List > 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 5698 | . . 3 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷) | |
| 2 | 1 | eleq2i 2833 | . 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 5993 | . . . . . . 7 ⊢ (𝐹 ↾ 𝐷) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷) |
| 6 | resmpt 6055 | . . . . . . 7 ⊢ (𝐷 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵)) | |
| 7 | 5, 6 | eqtrid 2789 | . . . . . 6 ⊢ (𝐷 ⊆ 𝐴 → (𝐹 ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ 𝐵)) |
| 8 | 7 | rneqd 5949 | . . . . 5 ⊢ (𝐷 ⊆ 𝐴 → ran (𝐹 ↾ 𝐷) = ran (𝑥 ∈ 𝐷 ↦ 𝐵)) |
| 9 | 8 | eleq2d 2827 | . . . 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 2737 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
| 13 | 12 | elrnmpt 5969 | . . . 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)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 ↦ cmpt 5225 ran crn 5686 ↾ cres 5687 “ cima 5688 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: reprpmtf1o 34641 ellcsrspsn 35646 |
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