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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatco | Structured version Visualization version GIF version |
Description: The predicate "defined at" for a function composition. (Contributed by AV, 8-Sep-2022.) |
Ref | Expression |
---|---|
dfatco | ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝐹 ∘ 𝐺) defAt 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfatcolem 44448 | . . . 4 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦) | |
2 | euex 2577 | . . . 4 ⊢ (∃!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦 → ∃𝑦 𝑋(𝐹 ∘ 𝐺)𝑦) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ∃𝑦 𝑋(𝐹 ∘ 𝐺)𝑦) |
4 | df-dm 5576 | . . . . 5 ⊢ dom (𝐹 ∘ 𝐺) = {𝑥 ∣ ∃𝑦 𝑥(𝐹 ∘ 𝐺)𝑦} | |
5 | 4 | eleq2i 2830 | . . . 4 ⊢ (𝑋 ∈ dom (𝐹 ∘ 𝐺) ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑦 𝑥(𝐹 ∘ 𝐺)𝑦}) |
6 | df-dfat 44312 | . . . . . . 7 ⊢ (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋}))) | |
7 | 6 | simplbi 501 | . . . . . 6 ⊢ (𝐺 defAt 𝑋 → 𝑋 ∈ dom 𝐺) |
8 | 7 | adantr 484 | . . . . 5 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → 𝑋 ∈ dom 𝐺) |
9 | breq1 5071 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥(𝐹 ∘ 𝐺)𝑦 ↔ 𝑋(𝐹 ∘ 𝐺)𝑦)) | |
10 | 9 | exbidv 1929 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑦 𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∃𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)) |
11 | 10 | elabg 3598 | . . . . 5 ⊢ (𝑋 ∈ dom 𝐺 → (𝑋 ∈ {𝑥 ∣ ∃𝑦 𝑥(𝐹 ∘ 𝐺)𝑦} ↔ ∃𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)) |
12 | 8, 11 | syl 17 | . . . 4 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝑋 ∈ {𝑥 ∣ ∃𝑦 𝑥(𝐹 ∘ 𝐺)𝑦} ↔ ∃𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)) |
13 | 5, 12 | syl5bb 286 | . . 3 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝑋 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)) |
14 | 3, 13 | mpbird 260 | . 2 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → 𝑋 ∈ dom (𝐹 ∘ 𝐺)) |
15 | dfdfat2 44321 | . 2 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ ∃!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)) | |
16 | 14, 1, 15 | sylanbrc 586 | 1 ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝐹 ∘ 𝐺) defAt 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2111 ∃!weu 2568 {cab 2715 {csn 4556 class class class wbr 5068 dom cdm 5566 ↾ cres 5568 ∘ ccom 5570 Fun wfun 6392 defAt wdfat 44309 ''''cafv2 44401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-res 5578 df-iota 6356 df-fun 6400 df-fn 6401 df-dfat 44312 df-afv2 44402 |
This theorem is referenced by: afv2co2 44450 |
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