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Mirrors > Home > MPE Home > Th. List > cores | Structured version Visualization version GIF version |
Description: Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cores | ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3451 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
2 | vex 3451 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brelrn 5901 | . . . . . 6 ⊢ (𝑧𝐵𝑦 → 𝑦 ∈ ran 𝐵) |
4 | ssel 3941 | . . . . . 6 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ 𝐶)) | |
5 | vex 3451 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | 5 | brresi 5950 | . . . . . . 7 ⊢ (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦𝐴𝑥)) |
7 | 6 | baib 537 | . . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥)) |
8 | 3, 4, 7 | syl56 36 | . . . . 5 ⊢ (ran 𝐵 ⊆ 𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴 ↾ 𝐶)𝑥 ↔ 𝑦𝐴𝑥))) |
9 | 8 | pm5.32d 578 | . . . 4 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
10 | 9 | exbidv 1925 | . . 3 ⊢ (ran 𝐵 ⊆ 𝐶 → (∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))) |
11 | 10 | opabbidv 5175 | . 2 ⊢ (ran 𝐵 ⊆ 𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)}) |
12 | df-co 5646 | . 2 ⊢ ((𝐴 ↾ 𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦(𝐴 ↾ 𝐶)𝑥)} | |
13 | df-co 5646 | . 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)} | |
14 | 11, 12, 13 | 3eqtr4g 2798 | 1 ⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ⊆ wss 3914 class class class wbr 5109 {copab 5171 ran crn 5638 ↾ cres 5639 ∘ ccom 5641 |
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-co 5646 df-dm 5647 df-rn 5648 df-res 5649 |
This theorem is referenced by: cocnvcnv1 6213 cores2 6215 relcoi2 6233 funresfunco 6546 fco2 6699 fcoi2 6721 domss2 9086 cottrcl 9663 canthp1lem2 10597 imasdsval2 17406 frmdss2 18681 gsumval3lem1 19690 gsumzres 19694 gsumzaddlem 19706 dprdf1 19820 kgencn2 22931 tsmsf1o 23519 lgamcvg2 26427 hhssims 30265 symgcom 31990 cycpmconjslem1 32059 cycpmconjslem2 32060 eulerpartgbij 33036 cvmlift2lem9a 33961 poimirlem9 36137 fourierdlem53 44490 |
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