![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > divsfval | Structured version Visualization version GIF version |
Description: Value of the function in qusval 17490. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.) |
Ref | Expression |
---|---|
ercpbl.r | ⊢ (𝜑 → ∼ Er 𝑉) |
ercpbl.v | ⊢ (𝜑 → 𝑉 ∈ 𝑊) |
ercpbl.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) |
Ref | Expression |
---|---|
divsfval | ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ercpbl.v | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊) | |
2 | ercpbl.r | . . . . . 6 ⊢ (𝜑 → ∼ Er 𝑉) | |
3 | 2 | ecss 8751 | . . . . 5 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉) |
4 | 1, 3 | ssexd 5324 | . . . 4 ⊢ (𝜑 → [𝐴] ∼ ∈ V) |
5 | eceq1 8743 | . . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ ) | |
6 | ercpbl.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
7 | 5, 6 | fvmptg 6996 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ [𝐴] ∼ ∈ V) → (𝐹‘𝐴) = [𝐴] ∼ ) |
8 | 4, 7 | sylan2 593 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = [𝐴] ∼ ) |
9 | 8 | expcom 414 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ )) |
10 | 6 | dmeqi 5904 | . . . . . . . 8 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) |
11 | 2 | ecss 8751 | . . . . . . . . . . 11 ⊢ (𝜑 → [𝑥] ∼ ⊆ 𝑉) |
12 | 1, 11 | ssexd 5324 | . . . . . . . . . 10 ⊢ (𝜑 → [𝑥] ∼ ∈ V) |
13 | 12 | ralrimivw 3150 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V) |
14 | dmmptg 6241 | . . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉) | |
15 | 13, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉) |
16 | 10, 15 | eqtrid 2784 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝑉) |
17 | 16 | eleq2d 2819 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝑉)) |
18 | 17 | notbid 317 | . . . . 5 ⊢ (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴 ∈ 𝑉)) |
19 | ndmfv 6926 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
20 | 18, 19 | syl6bir 253 | . . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∅)) |
21 | ecdmn0 8752 | . . . . . 6 ⊢ (𝐴 ∈ dom ∼ ↔ [𝐴] ∼ ≠ ∅) | |
22 | erdm 8715 | . . . . . . . . 9 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉) | |
23 | 2, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → dom ∼ = 𝑉) |
24 | 23 | eleq2d 2819 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ dom ∼ ↔ 𝐴 ∈ 𝑉)) |
25 | 24 | biimpd 228 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom ∼ → 𝐴 ∈ 𝑉)) |
26 | 21, 25 | biimtrrid 242 | . . . . 5 ⊢ (𝜑 → ([𝐴] ∼ ≠ ∅ → 𝐴 ∈ 𝑉)) |
27 | 26 | necon1bd 2958 | . . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → [𝐴] ∼ = ∅)) |
28 | 20, 27 | jcad 513 | . . 3 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → ((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅))) |
29 | eqtr3 2758 | . . 3 ⊢ (((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅) → (𝐹‘𝐴) = [𝐴] ∼ ) | |
30 | 28, 29 | syl6 35 | . 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ )) |
31 | 9, 30 | pm2.61d 179 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 Vcvv 3474 ∅c0 4322 ↦ cmpt 5231 dom cdm 5676 ‘cfv 6543 Er wer 8702 [cec 8703 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-er 8705 df-ec 8707 |
This theorem is referenced by: ercpbllem 17496 qusaddvallem 17499 qusgrp2 18943 frgpmhm 19635 frgpup3lem 19647 qusring2 20151 qusrhm 20880 |
Copyright terms: Public domain | W3C validator |