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Mirrors > Home > MPE Home > Th. List > dfima2 | Structured version Visualization version GIF version |
Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
Ref | Expression |
---|---|
dfima2 | ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5650 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | dfrn2 5848 | . 2 ⊢ ran (𝐴 ↾ 𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦} | |
3 | brres 5948 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦))) | |
4 | 3 | elv 3453 | . . . . 5 ⊢ (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) |
5 | 4 | exbii 1851 | . . . 4 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) |
6 | df-rex 3071 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) | |
7 | 5, 6 | bitr4i 278 | . . 3 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦) |
8 | 7 | abbii 2803 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
9 | 1, 2, 8 | 3eqtri 2765 | 1 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ∃wrex 3070 Vcvv 3447 class class class wbr 5109 ran crn 5638 ↾ cres 5639 “ cima 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 |
This theorem is referenced by: dfima3 6020 elimag 6021 imasng 6039 funimaexg 6591 dfimafn 6909 isoini 7287 dffin1-5 10332 dfimafnf 31603 ofpreima 31634 dfaimafn 45487 |
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